For all values of x, (3x - 5)^2 - (3x + 5)^2 = kx, where k is... | SAT Equivalent expressions Question, Answer Explained

Math · Advanced MathHard

For all values of $x$ , $(3x - 5)^2 - (3x + 5)^2 = kx$ , where $k$ is a constant. What is the value of $k$ ?

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Answer: -60

Expand each square: $(3x - 5)^2 = 9x^2 - 30x + 25$ and $(3x + 5)^2 = 9x^2 + 30x + 25$ . Subtracting gives $-30x - 30x = -60x$ , so $k = -60$ . The difference of squares gives the same result: $\left((3x - 5) - (3x + 5)\right)\left((3x - 5) + (3x + 5)\right) = (-10)(6x) = -60x$ .

Leaving out the middle terms when squaring gives $(9x^2 + 25) - (9x^2 + 25) = 0$ , which suggests $k = 0$ .

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