The equation above is true for all , where and are constants. What is the value of ?

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SAT Equivalent Expressions Question 198 | Free SAT Prep | Aniko

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Question 198·Hard·Equivalent Expressions

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\frac{2x-7}{x^2-5x+6}=\frac{A}{x-2}+\frac{B}{x-3}

The equation above is true for all $x\neq2,3$ , where $A$ and $B$ are constants. What is the value of $AB$ ?

For partial-fraction-style equivalence questions, start by factoring the denominator and multiplying through to clear fractions so you get a simple polynomial identity. Then use smart substitutions: pick values of $x$ that zero out one term at a time (like the roots of the denominator factors) to solve quickly for the unknown constants without heavy algebra. Finally, pay attention to what the question actually asks for (here, the product $AB$ ) so you do not stop after finding just $A$ and $B$ .

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Rewrite the denominator

Look at the denominator $x^2 - 5x + 6$ . Can you factor this quadratic into two binomials of the form $(x - m)(x - n)$ ?

Clear the fractions

Once you factor the denominator as $(x-2)(x-3)$ , multiply both sides of the equation by $(x-2)(x-3)$ so that you are working with a simpler polynomial equation instead of fractions.

Choose smart x-values

After you have an equation like $2x - 7 = A(x-3) + B(x-2)$ , try plugging in values of $x$ that make one of the terms on the right side equal to zero (think about $x=2$ and $x=3$ ). This will let you solve quickly for $A$ and $B$ .

Finish with the product

Once you know $A$ and $B$ , remember that the question asks for $AB$ , not $A$ or $B$ separately. Multiply your two values together at the end.

Desmos Guide

Graph both sides with sliders for A and B

In Desmos, enter the left side as

y1 = (2x - 7) / (x^2 - 5x + 6)

Then enter the right side as

y2 = A / (x - 2) + B / (x - 3)

Desmos will create sliders for $A$ and $B$ .

Match the graphs to determine A and B

Adjust the sliders for $A$ and $B$ until the graph of $y2$ exactly overlaps the graph of $y1$ (away from $x=2$ and $x=3$ , where the function is undefined). When the two graphs coincide everywhere else, note the values of $A$ and $B$ shown on the sliders.

Find AB from the slider values

Take the $A$ and $B$ values you read from the sliders and multiply them (you can type A*B into Desmos or calculate it yourself). The resulting number is the value of $AB$ you should choose in the answer options.

Step-by-step Explanation

Factor the denominator and clear the fractions

First factor the quadratic denominator.

The denominator is $x^2 - 5x + 6$ , which factors as $(x-2)(x-3)$ .

Multiply both sides of the original equation by $(x-2)(x-3)$ to remove the fractions:

2x - 7 = A(x-3) + B(x-2).

Now you have a polynomial equation that must hold for all $x$ .

Use convenient x-values to find A and B

Because the equation

2x - 7 = A(x-3) + B(x-2)

is true for all $x$ , you can choose values of $x$ that make one term on the right-hand side disappear.

Let $x = 2$ :
- Left side: $2(2) - 7 = 4 - 7 = -3$ .
- Right side: $A(2-3) + B(2-2) = A(-1) + B(0) = -A$ .
- So $-3 = -A$ , which gives $A = 3$ .
Let $x = 3$ :
- Left side: $2(3) - 7 = 6 - 7 = -1$ .
- Right side: $A(3-3) + B(3-2) = A(0) + B(1) = B$ .
- So $-1 = B$ , which gives $B = -1$ .

Now you know the specific values of $A$ and $B$ .

Compute the product AB

You found that $A = 3$ and $B = -1$ .

Now multiply them to get $AB$ :

AB = 3 \cdot (-1) = -3.

So, the value of $AB$ is $-3$ . This corresponds to choice B.

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