💡First, I need to determine the function ( f(x) ) given that ( f(x+1) = x^2 - 1 ).I'll start by expressing ( f(x+1) ) in terms of ( (x+1) ):[f(x+1) = (x+1)^2 - 1]Expanding ( (x+1)^2 ):[(x+1)^2 = x^2 + 2x + 1]Substituting back:[f(x+1) = x^2 + 2x + 1 - 1 = x^2 + 2x]Now, to find ( f(x) ), I'll replace ( x ) with ( x - 1 ) in the equation:[f(x) = (x - 1)^2 + 2(x - 1)]Expanding ( (x - 1)^2 ):[(x - 1)^2 = x^2 - 2x + 1]Substituting back:[f(x) = x^2 - 2x + 1 + 2x - 2 = x^2 - 2x - 1]However, this result doesn't match any of the provided options. It seems there might be an error in the substitution step. Let me re-examine the process.Starting again, from:[f(x+1) = x^2 + 2x]To find ( f(x) ), I'll replace ( x ) with ( x - 1 ):[f(x) = (x - 1)^2 + 2(x - 1)]Expanding:[(x - 1)^2 = x^2 - 2x + 1][2(x - 1) = 2x - 2]Combining these:[f(x) = x^2 - 2x + 1 + 2x - 2 = x^2 - 1]This still doesn't match the given options. It appears there's a mistake in the initial assumption. Let me try a different approach.Assume ( f(x) = x^2 + bx + c ). Then:[f(x+1) = (x+1)^2 + b(x+1) + c = x^2 + 2x + 1 + bx + b + c]Given ( f(x+1) = x^2 - 1 ), set the coefficients equal:[x^2 + (2 + b)x + (1 + b + c) = x^2 - 1]This gives the system:[2 + b = 0 quad Rightarrow quad b = -2][1 + b + c = -1 quad Rightarrow quad 1 - 2 + c = -1 quad Rightarrow quad c = 0]Thus, ( f(x) = x^2 - 2x ), which corresponds to option A.

💡First, I notice that the expression involves the difference of squares, which can be simplified using the formula (a^2 - b^2 = (a + b)(a - b)).Applying this formula to (275^2 - 245^2), I rewrite it as ((275 + 245)(275 - 245)).Next, I calculate the sum and difference of the numbers:- (275 + 245 = 520)- (275 - 245 = 30)Substituting these values back into the expression, I get:[frac{520 times 30}{30}]I observe that the (30) in the numerator and the denominator cancel each other out, leaving me with:[520]Therefore, the final answer is (520).

💡First, I need to understand what one-eighth of (2^{36}) means in terms of exponents. One-eighth can be expressed as (2^{-3}).Next, I'll apply the property of exponents that states (frac{a^m}{a^n} = a^{m-n}). Using this property, I can simplify (frac{2^{36}}{2^3}) to (2^{36-3}), which equals (2^{33}).Finally, since (2^{33}) is equal to (2^y), it follows that (y = 33).

💡Okay, so I have this problem here: I need to find the greatest integer value of ( b ) such that the expression ( frac{9x^3 + 4x^2 + 11x + 7}{x^2 + bx + 12} ) has a domain of all real numbers. Hmm, let me think about how to approach this.First, I remember that the domain of a rational expression like this is all real numbers except where the denominator equals zero. So, for the domain to be all real numbers, the denominator must never be zero for any real ( x ). That means the quadratic in the denominator, ( x^2 + bx + 12 ), must not have any real roots.Alright, so I need to make sure that the quadratic equation ( x^2 + bx + 12 = 0 ) has no real solutions. I recall that for a quadratic equation ( ax^2 + bx + c = 0 ), the discriminant ( D ) is given by ( D = b^2 - 4ac ). If the discriminant is negative, the quadratic has no real roots. So, I need to calculate the discriminant of the denominator and set it to be less than zero.Let me write that down:The discriminant ( D ) of ( x^2 + bx + 12 ) is:[ D = b^2 - 4 times 1 times 12 ][ D = b^2 - 48 ]For no real roots, we need:[ b^2 - 48 < 0 ][ b^2 < 48 ]So, ( b ) must satisfy ( b^2 < 48 ). To find the greatest integer value of ( b ), I need to find the largest integer less than ( sqrt{48} ).Calculating ( sqrt{48} ), I know that ( 6^2 = 36 ) and ( 7^2 = 49 ). Since ( 48 ) is between ( 36 ) and ( 49 ), ( sqrt{48} ) is between ( 6 ) and ( 7 ). Let me get a more precise estimate.( 6.9^2 = 47.61 ), which is less than ( 48 ).( 6.93^2 ) is approximately ( 48 ). So, ( sqrt{48} ) is approximately ( 6.93 ).Therefore, the greatest integer ( b ) such that ( b^2 < 48 ) is ( 6 ), because ( 7^2 = 49 ) is greater than ( 48 ).Wait, let me double-check that. If ( b = 6 ), then ( b^2 = 36 ), which is indeed less than ( 48 ). If ( b = 7 ), ( b^2 = 49 ), which is greater than ( 48 ), so ( 7 ) is too big. So, ( 6 ) is the largest integer that satisfies ( b^2 < 48 ).Therefore, the greatest integer value of ( b ) is ( 6 ).But just to make sure I didn't make any mistakes, let me recap:1. The denominator is ( x^2 + bx + 12 ).2. For it to have no real roots, discriminant must be negative.3. Discriminant ( D = b^2 - 48 ).4. So, ( b^2 < 48 ).5. ( sqrt{48} approx 6.93 ), so the greatest integer less than that is ( 6 ).Yes, that seems correct. I don't think I missed anything here. So, I'm confident that ( 6 ) is the right answer.**Final Answer**The greatest integer value of ( b ) is boxed{6}.

💡Okay, so I have this triangle problem here, and I need to find the length of AX. The problem says that CX bisects angle ACB. Hmm, angle bisector... I remember there's something called the Angle Bisector Theorem that might help here. Let me try to recall what that is.From what I remember, the Angle Bisector Theorem states that if a bisector of an angle in a triangle divides the opposite side into two segments, then those segments are proportional to the adjacent sides. So, in simpler terms, if we have a triangle ABC, and a bisector from angle C meets side AB at point X, then the ratio of AX to XB is equal to the ratio of AC to CB.Let me write that down to make it clearer. If CX bisects angle ACB, then:[frac{AX}{XB} = frac{AC}{CB}]Okay, so I need to find AX. The problem gives me some lengths: AC is 27, BC is 36, and BX is 32. Wait, so BX is part of side AB, right? So, if I can figure out the ratio, I can find AX.Let me plug in the values I have into the ratio. So, according to the theorem:[frac{AX}{32} = frac{27}{36}]Hmm, let me simplify that fraction on the right side. 27 divided by 36... both are divisible by 9. So, 27 divided by 9 is 3, and 36 divided by 9 is 4. So, that simplifies to 3/4.So now, the equation is:[frac{AX}{32} = frac{3}{4}]I need to solve for AX. To do that, I can cross-multiply. So, multiplying both sides by 32 gives:[AX = 32 times frac{3}{4}]Let me compute that. 32 divided by 4 is 8, and 8 multiplied by 3 is 24. So, AX is 24.Wait, let me double-check that to make sure I didn't make a mistake. So, if AX is 24 and XB is 32, then the ratio AX/XB is 24/32, which simplifies to 3/4. And AC/CB is 27/36, which also simplifies to 3/4. So, the ratios are equal, which makes sense according to the Angle Bisector Theorem.Therefore, I think I did this correctly. AX should be 24 units long.**Final Answer**The length of (AX) is boxed{24}.

💡First, I need to determine how many pencils each child has. There are a total of 18 pencils and 9 children.To find the number of pencils per child, I will divide the total number of pencils by the number of children.So, 18 pencils divided by 9 children equals 2 pencils per child.Therefore, each child has 2 pencils.