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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Medium

$- 4 x^{2} - 7 x = - 36$

What is the positive solution to the given equation?

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Explanation

Choice B is correct. Multiplying each side of the given equation by $negative 16$ yields $64 x^{2} + 112 x = 576$ . To complete the square, adding $49$ to each side of this equation yields $64 x^{2} + 112 x + 49 = 576 + 49$ , or ${(8 x + 7)}^{2} = 625$ . Taking the square root of each side of this equation yields two equations: $8 x + 7 = 25$ and $8 x + 7 = - 25$ . Subtracting $7$ from each side of the equation $8 x + 7 = 25$ yields $8 x equals 18$ . Dividing each side of this equation by $8$ yields $x = \frac{18}{8}$ , or $x = \frac{9}{4}$ . Therefore, $\frac{9}{4}$ is a solution to the given equation. Subtracting $7$ from each side of the equation $8 x + 7 = - 25$ yields $8 x = - 32$ . Dividing each side of this equation by $8$ yields $x = - 4$ . Therefore, the given equation has two solutions, $\frac{9}{4}$ and $negative 4$ . Since $\frac{9}{4}$ is positive, it follows that $\frac{9}{4}$ is the positive solution to the given equation.

Alternate approach: Adding $4 x squared$ and $7 x$ to each side of the given equation yields $0 = 4 x^{2} + 7 x - 36$ . The right-hand side of this equation can be rewritten as $4 x^{2} + 16 x - 9 x - 36$ . Factoring out the common factor of $4 x$ from the first two terms of this expression and the common factor of $negative 9$ from the second two terms yields $4 x (x + 4) - 9 (x + 4)$ . Factoring out the common factor of $left parenthesis x plus 4 right parenthesis$ from these two terms yields the expression $(4 x - 9) (x + 4)$ . Since this expression is equal to $0$ , it follows that either $4 x - 9 = 0$ or $x + 4 = 0$ . Adding $9$ to each side of the equation $4 x - 9 = 0$ yields $4 x equals 9$ . Dividing each side of this equation by $4$ yields $x = \frac{9}{4}$ . Therefore, $\frac{9}{4}$ is a positive solution to the given equation. Subtracting $4$ from each side of the equation $x + 4 = 0$ yields $x = - 4$ . Therefore, the given equation has two solutions, $\frac{9}{4}$ and $negative 4$ . Since $\frac{9}{4}$ is positive, it follows that $\frac{9}{4}$ is the positive solution to the given equation.

Choice A is incorrect. Substituting $\frac{7}{4}$ for $x$ in the given equation yields $- \frac{49}{2} = - 36$ , which is false.

Choice C is incorrect. Substituting $4$ for $x$ in the given equation yields $- 92 = - 36$ , which is false.

Choice D is incorrect. Substituting $7$ for $x$ in the given equation yields $- 245 = - 36$ , which is false.