💡Okay, so I have this equation: (7x)/4 + 2 = 6/x. I need to find the largest possible value of x in the form (a + b√c)/d, where a, b, c, and d are integers. Then, I have to calculate (acd)/b. Hmm, let me think about how to approach this step by step.First, I should probably get rid of the fractions to make the equation easier to handle. The equation has denominators 4 and x, so maybe I can multiply both sides by 4x to eliminate them. Let me try that.Multiplying every term by 4x:4x * (7x)/4 + 4x * 2 = 4x * (6/x)Simplifying each term:The first term: 4x * (7x)/4 = 7x^2 because the 4 cancels out.The second term: 4x * 2 = 8x.The third term: 4x * (6/x) = 24 because the x cancels out.So now the equation becomes:7x^2 + 8x = 24Hmm, that's a quadratic equation. I should bring all terms to one side to set it equal to zero. Let me subtract 24 from both sides:7x^2 + 8x - 24 = 0Alright, now I have a standard quadratic equation: 7x^2 + 8x - 24 = 0. To find the values of x, I can use the quadratic formula, which is x = [-b ± √(b² - 4ac)] / (2a). In this equation, a = 7, b = 8, and c = -24.Let me plug these values into the quadratic formula:x = [-8 ± √(8² - 4*7*(-24))] / (2*7)Calculating the discriminant first: 8² is 64, and 4*7*24 is... let me compute that. 4*7 is 28, and 28*24. Hmm, 28*20 is 560, and 28*4 is 112, so 560 + 112 = 672. But since c is -24, it's -4*7*(-24) which is +672. So the discriminant is 64 + 672 = 736.So now, x = [-8 ± √736] / 14Wait, √736 can be simplified, right? Let me see. 736 divided by 16 is 46, because 16*46 is 736. So √736 = √(16*46) = 4√46.So substituting back, we have:x = [-8 ± 4√46] / 14I can factor out a 4 from the numerator:x = [4*(-2 ± √46)] / 14Simplify the fraction by dividing numerator and denominator by 2:x = [2*(-2 ± √46)] / 7So, x = (-4 ± 2√46)/7Wait, hold on, let me check my steps again because I think I might have made a mistake in factoring.Starting from x = [-8 ± 4√46]/14I can factor out a 2 from numerator and denominator:x = [2*(-4 ± 2√46)] / 14Wait, no, that's not correct. Let me see:Wait, -8 is equal to 4*(-2), and 4√46 is 4*√46, so:x = [4*(-2 ± √46)] / 14Then, divide numerator and denominator by 2:x = [2*(-2 ± √46)] / 7Which is x = (-4 ± 2√46)/7Wait, that seems correct. Alternatively, I could have simplified earlier:x = [-8 ± 4√46]/14Divide numerator and denominator by 2:x = [-4 ± 2√46]/7Yes, that's correct.So, the two solutions are:x = (-4 + 2√46)/7 and x = (-4 - 2√46)/7Since we're looking for the largest possible value of x, we'll take the positive square root, so x = (-4 + 2√46)/7.But wait, the problem asks for the form (a + b√c)/d. Let me write it as:x = (-4 + 2√46)/7So, in this form, a = -4, b = 2, c = 46, and d = 7.Now, the problem asks for (acd)/b. Let me compute that:First, multiply a, c, d: (-4) * 46 * 7Compute 46 * 7 first: 46*7 is 322.Then, multiply by -4: -4 * 322 = -1288Now, divide by b, which is 2: -1288 / 2 = -644Wait, that's different from the previous answer. Did I make a mistake somewhere?Wait, let me check my steps again.Starting from the quadratic equation:7x^2 + 8x - 24 = 0Using quadratic formula:x = [-8 ± √(64 + 672)] / 14√736 = 4√46, so x = [-8 ± 4√46]/14Simplify numerator and denominator by dividing by 2:x = [-4 ± 2√46]/7So, x = (-4 + 2√46)/7 is the positive solution.So, a = -4, b = 2, c = 46, d = 7.Compute (a*c*d)/b:(-4)*46*7 / 2First, compute 46*7: 46*7=322Then, multiply by -4: -4*322 = -1288Divide by 2: -1288 / 2 = -644Wait, but in the initial solution, the assistant got -322. So, which one is correct?Wait, maybe I made a mistake in simplifying the quadratic formula.Let me go back.Original equation: (7x)/4 + 2 = 6/xMultiply both sides by 4x:7x^2 + 8x = 247x^2 + 8x -24 = 0Quadratic formula: x = [-8 ± √(64 + 4*7*24)] / (2*7)Compute discriminant: 64 + 4*7*244*7=28, 28*24=672, so discriminant is 64+672=736√736=√(16*46)=4√46Thus, x = [-8 ±4√46]/14Simplify numerator and denominator by dividing by 2:x = [-4 ±2√46]/7So, positive solution is x = (-4 + 2√46)/7Thus, a=-4, b=2, c=46, d=7Compute (a*c*d)/b: (-4)*46*7 /2Calculate step by step:First, 46*7=322Then, 322*(-4)= -1288Then, -1288 /2= -644So, the correct value is -644, not -322.Wait, so why did the initial assistant get -322? Let me check their steps.They had x = (-4 +4√46)/7, so a=-4, b=4, c=46, d=7Then, (a*c*d)/b = (-4*46*7)/4 = (-4*46*7)/4 = (-4/4)*46*7= -1*46*7= -322Ah, I see. So, in their solution, they had b=4, but in my solution, I have b=2.Wait, why the discrepancy? Let me check.In my solution, I simplified x = [-8 ±4√46]/14 to x = [-4 ±2√46]/7 by dividing numerator and denominator by 2.But in the assistant's solution, they didn't simplify further and kept it as x = (-8 +8√46)/14, which simplifies to (-4 +4√46)/7.Wait, that's different. So, in the assistant's solution, they had:x = (-8 ±8√46)/14 = (-4 ±4√46)/7But in my solution, I simplified further by dividing numerator and denominator by 2, getting x = (-4 ±2√46)/7.Wait, but that would mean that the assistant's solution is incorrect because they didn't simplify correctly.Wait, let me check:From x = [-8 ±4√46]/14Divide numerator and denominator by 2:Numerator: (-8)/2 = -4, (4√46)/2=2√46Denominator:14/2=7Thus, x = (-4 ±2√46)/7So, the correct simplified form is x = (-4 ±2√46)/7, meaning a=-4, b=2, c=46, d=7.Therefore, the assistant's solution was incorrect because they didn't simplify the fraction correctly, leading to b=4 instead of b=2.Thus, the correct value of (acd)/b is (-4*46*7)/2 = (-4*46*7)/2 = (-4/2)*46*7 = (-2)*46*7 = -644.Wait, but let me double-check the simplification.Original expression after quadratic formula: x = [-8 ±4√46]/14Factor out 2 from numerator: 2*(-4 ±2√46)/14Simplify denominator: 14=2*7, so 2 cancels out, leaving (-4 ±2√46)/7Yes, that's correct.Therefore, the correct values are a=-4, b=2, c=46, d=7, leading to (acd)/b = (-4*46*7)/2 = -644.So, the initial assistant's answer was incorrect because they didn't simplify the fraction properly, keeping b=4 instead of simplifying it to b=2.Therefore, the correct answer should be -644.

💡Okay, so I need to find the angle between two vectors. The vectors given are (begin{pmatrix} 3 -2 2 end{pmatrix}) and (begin{pmatrix} -2 2 1 end{pmatrix}). Hmm, I remember that to find the angle between two vectors, I can use the dot product formula. Let me try to recall how that works.First, I think the formula involves the dot product of the two vectors divided by the product of their magnitudes. Yeah, something like (cos theta = frac{mathbf{u} cdot mathbf{v}}{|mathbf{u}| |mathbf{v}|}). So, I need to compute the dot product of the two vectors and then divide it by the product of their magnitudes. Then, take the inverse cosine to find the angle in degrees.Alright, let's start with the dot product. The dot product of two vectors is calculated by multiplying their corresponding components and then adding them up. So, for vectors (mathbf{u} = begin{pmatrix} 3 -2 2 end{pmatrix}) and (mathbf{v} = begin{pmatrix} -2 2 1 end{pmatrix}), the dot product (mathbf{u} cdot mathbf{v}) should be:(3 times (-2) + (-2) times 2 + 2 times 1).Let me compute each term:- (3 times (-2) = -6)- (-2 times 2 = -4)- (2 times 1 = 2)Now, adding these up: (-6 + (-4) + 2 = -6 - 4 + 2 = -8). So, the dot product is (-8).Next, I need to find the magnitudes of both vectors. The magnitude of a vector (mathbf{u}) is given by (|mathbf{u}| = sqrt{u_1^2 + u_2^2 + u_3^2}).Let's compute the magnitude of (mathbf{u}):(|mathbf{u}| = sqrt{3^2 + (-2)^2 + 2^2} = sqrt{9 + 4 + 4}).Calculating inside the square root: (9 + 4 = 13), and (13 + 4 = 17). So, (|mathbf{u}| = sqrt{17}).Now, the magnitude of (mathbf{v}):(|mathbf{v}| = sqrt{(-2)^2 + 2^2 + 1^2} = sqrt{4 + 4 + 1}).Adding those up: (4 + 4 = 8), and (8 + 1 = 9). So, (|mathbf{v}| = sqrt{9} = 3).Alright, so now I have the dot product and the magnitudes. Let's plug these into the cosine formula:(cos theta = frac{-8}{sqrt{17} times 3}).Simplifying the denominator: (sqrt{17} times 3 = 3sqrt{17}).So, (cos theta = frac{-8}{3sqrt{17}}).Hmm, okay, so to find the angle (theta), I need to take the inverse cosine (arccos) of (frac{-8}{3sqrt{17}}).But wait, before I proceed, I should check if this value is within the domain of the arccos function. The domain is between -1 and 1. Let me compute the approximate value of (frac{-8}{3sqrt{17}}).First, calculate (sqrt{17}). I remember that (sqrt{16} = 4) and (sqrt{17}) is approximately 4.123.So, (3sqrt{17} approx 3 times 4.123 = 12.369).Then, (frac{-8}{12.369} approx -0.646).Since -0.646 is between -1 and 1, it's valid for the arccos function. Good.Now, let me compute (theta = cos^{-1}(-0.646)). I need to make sure my calculator is in degree mode because the question asks for the angle in degrees.Calculating (cos^{-1}(-0.646)). Hmm, I know that (cos(120^circ) = -0.5) and (cos(135^circ) = -frac{sqrt{2}}{2} approx -0.7071). Since -0.646 is between -0.5 and -0.7071, the angle should be between 120° and 135°.Let me use a calculator to find the exact value.[Imagining using a calculator]After calculating, I get approximately 130.5 degrees. Let me verify that.Wait, let me check: (cos(130.5^circ)) should be approximately -0.646.Using the calculator, (cos(130.5)) is indeed around -0.646. So, that seems correct.Alternatively, I can compute it more precisely.But for the purposes of this problem, I think 130.5 degrees is a reasonable approximation. However, since the problem might expect an exact expression or a more precise value, maybe I should express it in terms of inverse cosine.Wait, looking back at the initial problem, it just asks for the angle in degrees, so probably an approximate value is acceptable. But let me see if I can express it more accurately.Alternatively, maybe I can rationalize the denominator in the cosine expression to make it neater.So, (cos theta = frac{-8}{3sqrt{17}}). If I rationalize the denominator:Multiply numerator and denominator by (sqrt{17}):(cos theta = frac{-8sqrt{17}}{3 times 17} = frac{-8sqrt{17}}{51}).So, (cos theta = frac{-8sqrt{17}}{51}).But I don't think that helps much in terms of simplifying the angle. So, perhaps it's better to just compute the numerical value.Alternatively, maybe I can write the exact value as (cos^{-1}left(frac{-8}{3sqrt{17}}right)), but the question asks for the angle in degrees, so I think a numerical value is expected.Wait, but in the initial problem, the user wrote the same question twice, and the assistant provided an answer in terms of inverse cosine. Maybe they just wanted the expression, but the user is asking again, perhaps expecting a numerical value.Hmm, to be safe, I can provide both the exact expression and the approximate degree measure.But let me check my calculations again to make sure I didn't make any mistakes.First, the dot product: 3*(-2) = -6, (-2)*2 = -4, 2*1 = 2. Sum: -6 -4 +2 = -8. Correct.Magnitudes: For u, sqrt(3^2 + (-2)^2 + 2^2) = sqrt(9 +4 +4) = sqrt(17). Correct.For v, sqrt((-2)^2 + 2^2 +1^2) = sqrt(4 +4 +1) = sqrt(9) = 3. Correct.So, cos(theta) = -8/(sqrt(17)*3) = -8/(3sqrt(17)) ≈ -0.646. Correct.Then, arccos(-0.646) ≈ 130.5 degrees. Let me double-check with a calculator.Using calculator: arccos(-0.646). Let me compute it step by step.First, in radians, arccos(-0.646) is approximately 2.275 radians. To convert to degrees, multiply by (180/pi):2.275 * (180/3.1416) ≈ 2.275 * 57.2958 ≈ 130.5 degrees. Yes, that's correct.So, the angle is approximately 130.5 degrees.But, to be precise, let me compute it more accurately.Using a calculator, arccos(-0.646):First, find the reference angle: arccos(0.646) ≈ 50 degrees (since cos(50°) ≈ 0.6428, which is close to 0.646). So, the reference angle is approximately 50 degrees.Since cosine is negative, the angle is in the second quadrant, so 180° - 50° = 130°. But to get a more accurate value, let's compute it precisely.Using a calculator, arccos(-0.646) ≈ 130.49°, which is approximately 130.5°. So, 130.5 degrees is a good approximation.Alternatively, if I use more decimal places, say, -0.64605, the angle would be slightly more precise, but for most purposes, 130.5 degrees is sufficient.Therefore, the angle between the two vectors is approximately 130.5 degrees.Wait, just to make sure, let me check if I used the correct formula.Yes, the formula is (cos theta = frac{mathbf{u} cdot mathbf{v}}{|mathbf{u}| |mathbf{v}|}), which I used correctly.And the calculations for the dot product and magnitudes seem correct.So, I think my answer is accurate.

💡Okay, so I have this problem where I need to find the probability distribution of a random variable X, which represents the number of times a six appears when rolling a die three times. Then, I also need to compute the expected value, variance, and standard deviation of this random variable. Hmm, let me think about how to approach this step by step.First, I know that when dealing with probabilities involving multiple trials, especially with something like rolling a die, the binomial distribution might be useful here. The binomial distribution applies when there are a fixed number of independent trials, each with two possible outcomes (success or failure), and the probability of success is constant.In this case, rolling a die three times fits the bill because there are a fixed number of trials (three rolls), each roll is independent of the others, and each roll has two outcomes: either a six (which I'll consider a success) or not a six (a failure). The probability of rolling a six on a single roll is 1/6, and the probability of not rolling a six is 5/6.So, if I let X be the number of sixes in three rolls, X can take on the values 0, 1, 2, or 3. I need to find the probability for each of these outcomes.The formula for the binomial probability is:P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)Where:- C(n, k) is the combination of n things taken k at a time.- p is the probability of success.- n is the number of trials.- k is the number of successes.So, plugging in the values for our problem:- n = 3- p = 1/6- k = 0, 1, 2, 3Let me compute each probability one by one.1. For X = 0 (no sixes in three rolls):P(X = 0) = C(3, 0) * (1/6)^0 * (5/6)^3C(3, 0) is 1, (1/6)^0 is 1, and (5/6)^3 is 125/216.So, P(X = 0) = 1 * 1 * 125/216 = 125/216.2. For X = 1 (exactly one six in three rolls):P(X = 1) = C(3, 1) * (1/6)^1 * (5/6)^2C(3, 1) is 3, (1/6)^1 is 1/6, and (5/6)^2 is 25/36.So, P(X = 1) = 3 * (1/6) * (25/36) = 3 * (25/216) = 75/216.3. For X = 2 (exactly two sixes in three rolls):P(X = 2) = C(3, 2) * (1/6)^2 * (5/6)^1C(3, 2) is 3, (1/6)^2 is 1/36, and (5/6)^1 is 5/6.So, P(X = 2) = 3 * (1/36) * (5/6) = 3 * (5/216) = 15/216.4. For X = 3 (all three rolls are sixes):P(X = 3) = C(3, 3) * (1/6)^3 * (5/6)^0C(3, 3) is 1, (1/6)^3 is 1/216, and (5/6)^0 is 1.So, P(X = 3) = 1 * (1/216) * 1 = 1/216.Let me check if these probabilities add up to 1, as they should.125/216 + 75/216 + 15/216 + 1/216 = (125 + 75 + 15 + 1)/216 = 216/216 = 1.Good, that checks out. So, the probability distribution is:X | 0 | 1 | 2 | 3---|---|---|---|---P(X) | 125/216 | 75/216 | 15/216 | 1/216Now, moving on to the expected value, variance, and standard deviation.For a binomial distribution, the expected value (mean) is given by:E(X) = n * pPlugging in the values:E(X) = 3 * (1/6) = 3/6 = 1/2.So, the expected number of sixes in three rolls is 0.5.Next, the variance of a binomial distribution is given by:Var(X) = n * p * (1 - p)Plugging in the values:Var(X) = 3 * (1/6) * (5/6) = (3 * 5) / (6 * 6) = 15/36 = 5/12.To find the standard deviation, we take the square root of the variance:SD(X) = sqrt(Var(X)) = sqrt(5/12).Simplifying sqrt(5/12), we can write it as sqrt(15)/6 because sqrt(5/12) = sqrt(5)/sqrt(12) = sqrt(5)/(2*sqrt(3)) = (sqrt(5)*sqrt(3))/(2*3) = sqrt(15)/6.Let me recap:- The probability distribution of X is as above.- The expected value is 1/2.- The variance is 5/12.- The standard deviation is sqrt(15)/6.I think that covers everything the problem is asking for. I should double-check my calculations to make sure I didn't make any arithmetic errors.Checking the probabilities again:- 125/216 ≈ 0.5787- 75/216 ≈ 0.3472- 15/216 ≈ 0.0694- 1/216 ≈ 0.0046Adding those approximate decimals: 0.5787 + 0.3472 + 0.0694 + 0.0046 ≈ 1.0, which is correct.For the expected value:E(X) = 0*(125/216) + 1*(75/216) + 2*(15/216) + 3*(1/216)= 0 + 75/216 + 30/216 + 3/216= (75 + 30 + 3)/216= 108/216= 1/2.That's correct.For variance:E(X^2) = 0^2*(125/216) + 1^2*(75/216) + 2^2*(15/216) + 3^2*(1/216)= 0 + 75/216 + 4*(15/216) + 9*(1/216)= 75/216 + 60/216 + 9/216= (75 + 60 + 9)/216= 144/216= 2/3.Then, Var(X) = E(X^2) - [E(X)]^2 = 2/3 - (1/2)^2 = 2/3 - 1/4 = (8/12 - 3/12) = 5/12.That matches what I got earlier.And the standard deviation is sqrt(5/12), which is approximately 0.6455, but in exact terms, it's sqrt(15)/6.So, everything seems consistent. I think I've got it right.

💡First, I need to understand the given sets and the condition provided. Set ( A ) contains the elements ( 0 ) and ( a ), while set ( B ) contains ( 3^a ) and ( 1 ). The intersection of ( A ) and ( B ) is given as ( {1} ), which means that ( 1 ) is the only common element between the two sets.Since ( 1 ) must be in both sets, it implies that ( a ) in set ( A ) must be ( 1 ). This is because ( 0 ) cannot be equal to ( 1 ). Therefore, ( a = 1 ).With ( a = 1 ), set ( A ) becomes ( {0, 1} ). For set ( B ), substituting ( a = 1 ) gives ( 3^1 = 3 ), so set ( B ) is ( {3, 1} ).Now, to find the union of ( A ) and ( B ), I combine all the unique elements from both sets. Set ( A ) has ( 0 ) and ( 1 ), while set ( B ) has ( 3 ) and ( 1 ). Combining these, the union ( A cup B ) will be ( {0, 1, 3} ).Finally, I present the union of sets ( A ) and ( B ) as ( {0, 1, 3} ).

💡Okay, so I have this system of equations:left{begin{array}{l}a y + b x = c c x + a z = b b z + c y = aend{array}right.And I need to solve for x, y, and z. Hmm, let's see. I remember that solving a system of equations usually involves substitution or elimination. Maybe I can use elimination here because the equations look somewhat symmetrical.First, let me write down the equations again for clarity:1. ( a y + b x = c )2. ( c x + a z = b )3. ( b z + c y = a )Hmm, each equation has two variables multiplied by constants and equals another constant. It's a linear system, so maybe I can express each variable in terms of the others and substitute.Let me try to solve the first equation for y:From equation 1: ( a y = c - b x ) => ( y = frac{c - b x}{a} )Okay, so y is expressed in terms of x. Now, let's plug this expression for y into equation 3.Equation 3: ( b z + c y = a )Substituting y:( b z + c left( frac{c - b x}{a} right) = a )Simplify:( b z + frac{c^2 - b c x}{a} = a )Multiply both sides by a to eliminate the denominator:( a b z + c^2 - b c x = a^2 )Let me rearrange this:( a b z = a^2 - c^2 + b c x )So,( z = frac{a^2 - c^2 + b c x}{a b} )Simplify:( z = frac{a^2 - c^2}{a b} + frac{b c x}{a b} )Which simplifies to:( z = frac{a^2 - c^2}{a b} + frac{c x}{a} )Okay, so now I have z in terms of x. Let me keep that aside.Now, let's look at equation 2:( c x + a z = b )We can substitute z from above into this equation.So,( c x + a left( frac{a^2 - c^2}{a b} + frac{c x}{a} right) = b )Simplify term by term:First term: ( c x )Second term: ( a * frac{a^2 - c^2}{a b} = frac{a^2 - c^2}{b} )Third term: ( a * frac{c x}{a} = c x )So, putting it all together:( c x + frac{a^2 - c^2}{b} + c x = b )Combine like terms:( 2 c x + frac{a^2 - c^2}{b} = b )Let me solve for x:Subtract ( frac{a^2 - c^2}{b} ) from both sides:( 2 c x = b - frac{a^2 - c^2}{b} )Multiply both sides by b to eliminate the denominator:( 2 c x b = b^2 - (a^2 - c^2) )Simplify the right side:( 2 c x b = b^2 - a^2 + c^2 )So,( x = frac{b^2 - a^2 + c^2}{2 c b} )Which can be written as:( x = frac{b^2 + c^2 - a^2}{2 b c} )Okay, so I found x in terms of a, b, c. Now, let's find y and z.From earlier, we had:( y = frac{c - b x}{a} )Substitute x:( y = frac{c - b left( frac{b^2 + c^2 - a^2}{2 b c} right)}{a} )Simplify inside the numerator:( c - frac{b (b^2 + c^2 - a^2)}{2 b c} = c - frac{b^2 + c^2 - a^2}{2 c} )To combine terms, let's write c as ( frac{2 c^2}{2 c} ):( frac{2 c^2}{2 c} - frac{b^2 + c^2 - a^2}{2 c} = frac{2 c^2 - b^2 - c^2 + a^2}{2 c} )Simplify numerator:( 2 c^2 - c^2 = c^2 ), so:( frac{c^2 - b^2 + a^2}{2 c} )Thus,( y = frac{c^2 - b^2 + a^2}{2 c a} = frac{a^2 + c^2 - b^2}{2 a c} )Okay, so y is found.Now, let's find z. Earlier, we had:( z = frac{a^2 - c^2}{a b} + frac{c x}{a} )Substitute x:( z = frac{a^2 - c^2}{a b} + frac{c}{a} left( frac{b^2 + c^2 - a^2}{2 b c} right) )Simplify the second term:( frac{c}{a} * frac{b^2 + c^2 - a^2}{2 b c} = frac{b^2 + c^2 - a^2}{2 a b} )So, z becomes:( z = frac{a^2 - c^2}{a b} + frac{b^2 + c^2 - a^2}{2 a b} )Combine the fractions:Let me write both terms with denominator 2 a b:First term: ( frac{2 (a^2 - c^2)}{2 a b} )Second term: ( frac{b^2 + c^2 - a^2}{2 a b} )So,( z = frac{2 a^2 - 2 c^2 + b^2 + c^2 - a^2}{2 a b} )Simplify numerator:( 2 a^2 - a^2 = a^2 )( -2 c^2 + c^2 = -c^2 )So,( z = frac{a^2 - c^2 + b^2}{2 a b} = frac{a^2 + b^2 - c^2}{2 a b} )Alright, so z is found.So, summarizing:( x = frac{b^2 + c^2 - a^2}{2 b c} )( y = frac{a^2 + c^2 - b^2}{2 a c} )( z = frac{a^2 + b^2 - c^2}{2 a b} )Hmm, these expressions look familiar. They resemble the expressions for the cosines of angles in a triangle, based on the Law of Cosines. In the Law of Cosines, for a triangle with sides a, b, c opposite angles A, B, C respectively, we have:( cos A = frac{b^2 + c^2 - a^2}{2 b c} )Similarly for cos B and cos C.So, if we think of x, y, z as cosines of angles opposite sides a, b, c respectively, then this system of equations is essentially expressing the Law of Cosines for a triangle.Therefore, the geometric interpretation is that x, y, z represent the cosines of the angles opposite sides a, b, c in a triangle. This means that the system has a solution if and only if a, b, c can form a triangle, i.e., they satisfy the triangle inequality.So, in essence, solving this system gives the angles of a triangle with sides a, b, c, provided such a triangle exists.**Final Answer**The solution to the system is:boxed{x = frac{b^2 + c^2 - a^2}{2bc}}, quad boxed{y = frac{a^2 + c^2 - b^2}{2ac}}, quad boxed{z = frac{a^2 + b^2 - c^2}{2ab}}Geometrically, these represent the cosines of the angles opposite sides (a), (b), and (c) in a triangle.

💡Alright, so I have this problem here: If ( r ) and ( s ) are the roots of the quadratic equation ( x^2 - px - q = 0 ), find the value of ( r^3 + s^3 ). The options are A) ( p^3 - 3pq ), B) ( p^3 + 3pq ), C) ( p^3 - pq ), and D) ( 3p^2 - q^2 ).Okay, first things first, I remember that for a quadratic equation of the form ( x^2 + bx + c = 0 ), the sum of the roots is ( -b ) and the product is ( c ). So in this case, the equation is ( x^2 - px - q = 0 ). Comparing this to the standard form, ( b = -p ) and ( c = -q ). Therefore, the sum of the roots ( r + s = p ) and the product ( rs = -q ).Now, the question is asking for ( r^3 + s^3 ). Hmm, I think there's a formula for the sum of cubes. Let me recall... I believe it's ( r^3 + s^3 = (r + s)(r^2 - rs + s^2) ). Yeah, that sounds right. Alternatively, I also remember that ( r^3 + s^3 ) can be expressed in terms of ( (r + s)^3 ) and ( rs ). Let me see... Expanding ( (r + s)^3 ), we get ( r^3 + 3r^2s + 3rs^2 + s^3 ). So, ( (r + s)^3 = r^3 + s^3 + 3rs(r + s) ). Therefore, rearranging, ( r^3 + s^3 = (r + s)^3 - 3rs(r + s) ).Okay, so using that formula, let's plug in the values we have. We know ( r + s = p ) and ( rs = -q ). So substituting these into the formula, we get:( r^3 + s^3 = p^3 - 3(-q)(p) ).Simplifying that, the negative times negative becomes positive, so:( r^3 + s^3 = p^3 + 3pq ).Let me double-check to make sure I didn't make any mistakes. So, starting from the sum and product of roots, I used the identity for the sum of cubes, substituted the known values, and simplified. It seems correct. So, the answer should be ( p^3 + 3pq ), which is option B.Just to be thorough, let me consider if there's another way to approach this. Maybe by actually finding the roots and then cubing them? But that might be more complicated. The quadratic equation is ( x^2 - px - q = 0 ), so the roots can be found using the quadratic formula: ( x = frac{p pm sqrt{p^2 + 4q}}{2} ). Then, cubing each root and adding them would be quite involved. It might lead to the same result, but it's definitely more work. So, I think the method I used initially is the most straightforward and efficient.Another thing I could consider is plugging in some numbers to test. Let's say ( p = 2 ) and ( q = 1 ). Then the equation becomes ( x^2 - 2x - 1 = 0 ). The roots of this equation can be found using the quadratic formula: ( x = frac{2 pm sqrt{4 + 4}}{2} = frac{2 pm sqrt{8}}{2} = 1 pm sqrt{2} ). So, ( r = 1 + sqrt{2} ) and ( s = 1 - sqrt{2} ).Calculating ( r^3 + s^3 ):First, ( r^3 = (1 + sqrt{2})^3 ). Expanding this, ( (1 + sqrt{2})^3 = 1 + 3sqrt{2} + 3(2) + (sqrt{2})^3 = 1 + 3sqrt{2} + 6 + 2sqrt{2} = 7 + 5sqrt{2} ).Similarly, ( s^3 = (1 - sqrt{2})^3 = 1 - 3sqrt{2} + 3(2) - (sqrt{2})^3 = 1 - 3sqrt{2} + 6 - 2sqrt{2} = 7 - 5sqrt{2} ).Adding them together: ( r^3 + s^3 = (7 + 5sqrt{2}) + (7 - 5sqrt{2}) = 14 ).Now, let's compute ( p^3 + 3pq ) with ( p = 2 ) and ( q = 1 ): ( 2^3 + 3(2)(1) = 8 + 6 = 14 ). That matches the result from directly calculating ( r^3 + s^3 ). So, this confirms that the formula works correctly in this case.Just to be extra sure, let's try another set of numbers. Let me pick ( p = 3 ) and ( q = 2 ). Then the equation is ( x^2 - 3x - 2 = 0 ). The roots are ( x = frac{3 pm sqrt{9 + 8}}{2} = frac{3 pm sqrt{17}}{2} ). So, ( r = frac{3 + sqrt{17}}{2} ) and ( s = frac{3 - sqrt{17}}{2} ).Calculating ( r^3 + s^3 ):First, ( r^3 = left( frac{3 + sqrt{17}}{2} right)^3 ). Let's compute this:( left( frac{3 + sqrt{17}}{2} right)^3 = frac{(3 + sqrt{17})^3}{8} ).Expanding ( (3 + sqrt{17})^3 ):( = 3^3 + 3 cdot 3^2 cdot sqrt{17} + 3 cdot 3 cdot (sqrt{17})^2 + (sqrt{17})^3 )( = 27 + 27sqrt{17} + 3 cdot 3 cdot 17 + 17sqrt{17} )( = 27 + 27sqrt{17} + 153 + 17sqrt{17} )( = 180 + 44sqrt{17} ).So, ( r^3 = frac{180 + 44sqrt{17}}{8} = frac{45 + 11sqrt{17}}{2} ).Similarly, ( s^3 = left( frac{3 - sqrt{17}}{2} right)^3 = frac{(3 - sqrt{17})^3}{8} ).Expanding ( (3 - sqrt{17})^3 ):( = 3^3 - 3 cdot 3^2 cdot sqrt{17} + 3 cdot 3 cdot (sqrt{17})^2 - (sqrt{17})^3 )( = 27 - 27sqrt{17} + 153 - 17sqrt{17} )( = 180 - 44sqrt{17} ).So, ( s^3 = frac{180 - 44sqrt{17}}{8} = frac{45 - 11sqrt{17}}{2} ).Adding ( r^3 + s^3 ):( frac{45 + 11sqrt{17}}{2} + frac{45 - 11sqrt{17}}{2} = frac{90}{2} = 45 ).Now, let's compute ( p^3 + 3pq ) with ( p = 3 ) and ( q = 2 ): ( 3^3 + 3(3)(2) = 27 + 18 = 45 ). Again, this matches the result from directly calculating ( r^3 + s^3 ).This further solidifies that the formula ( r^3 + s^3 = p^3 + 3pq ) is correct.Just to cover all bases, let me think if there's any alternative method or identity that could be used here. I recall that sometimes, for expressions involving powers of roots, we can use recurrence relations or Newton's identities, but in this case, since it's only up to the third power, the direct approach using the sum and product of roots seems sufficient and straightforward.Another point to consider is whether the initial formula I used, ( r^3 + s^3 = (r + s)^3 - 3rs(r + s) ), is correct. Let me verify this by expanding ( (r + s)^3 ):( (r + s)^3 = r^3 + 3r^2s + 3rs^2 + s^3 ).If we subtract ( 3rs(r + s) ) from this, we get:( (r + s)^3 - 3rs(r + s) = r^3 + 3r^2s + 3rs^2 + s^3 - 3r^2s - 3rs^2 = r^3 + s^3 ).Yes, that checks out. So, the formula is indeed valid.In summary, by using Vieta's formulas to find the sum and product of the roots, and then applying the identity for the sum of cubes, we can efficiently compute ( r^3 + s^3 ) in terms of ( p ) and ( q ). The result is ( p^3 + 3pq ), which corresponds to option B.I think I've thoroughly explored this problem from different angles, verified the formula, and tested it with specific examples to ensure its correctness. I'm confident that the answer is indeed option B.