The expression 3x^2 + 7x - 6 can be written as (3x + a)(x +... | SAT Equivalent expressions Question, Answer Explained

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The expression $3x^2 + 7x - 6$ can be written as $(3x + a)(x + b)$ , where $a$ and $b$ are integers. What is the value of $b - a$ ?

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Answer: 5

Expanding $(3x + a)(x + b)$ gives $3x^2 + (3b + a)x + ab$ . So $ab = -6$ and $3b + a = 7$ . Testing factor pairs of $-6$ , the values $a = -2$ and $b = 3$ work, since $3(3) + (-2) = 7$ . So $3x^2 + 7x - 6 = (3x - 2)(x + 3)$ , and $b - a = 3 - (-2) = 5$ .

Placing the numbers the other way, $a = 3$ and $b = -2$ , gives a middle coefficient of $3(-2) + 3 = -3$ , not $7$ .

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