SAT Practice Test #3 Section 3: Math Test - No Calculator #13 - Free Printable

- Multiply both sides by $ax - 2$ (valid since $x \ne \frac{2}{a}$):
$$
24x^2 + 25x - 47 = (-8x - 3)(ax - 2) - 53
$$

- Expand the right-hand side:
$$
(-8x - 3)(ax - 2) = -8x(ax - 2) - 3(ax - 2) = -8a x^2 + 16x - 3a x + 6
$$
So the full right-hand side is:
$$
-8a x^2 + 16x - 3a x + 6 - 53 = -8a x^2 + (16 - 3a)x - 47
$$

- Now equate both sides:
$$
24x^2 + 25x - 47 = -8a x^2 + (16 - 3a)x - 47
$$

- Since this is true for all $x \ne \frac{2}{a}$, coefficients of corresponding powers must be equal.

- Compare $x^2$ terms:
$$
24 = -8a \Rightarrow a = -3
$$

- Verify with $x$ terms:
Left: $25$, Right: $16 - 3(-3) = 16 + 9 = 25$ — matches.

- Constant terms: both are $-47$ — matches.

- Thus, $a = -3$

B) -3