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

What is the positive solution to the given equation?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-57"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

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

What is the positive solution to the given equation?

","comment":{"@type":"Comment","text":"

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

Alternate approach: Adding $4x^{2}$ and $7x$ to each side of the given equation yields $(0=4x^{2}+7x-36)$. The right-hand side of this equation can be rewritten as $(4x^{2}+16x-9x-36)$. Factoring out the common factor of $4x$ from the first two terms of this expression and the common factor of $-9$ from the second two terms yields $(4x(x+4)-9(x+4))$. Factoring out the common factor of $(x+4)$ from these two terms yields the expression $((4x-9)(x+4))$. Since this expression is equal to $0$, it follows that either $(4x-9=0)$ or $\\bm{x}+4=0$. Adding $9$ to each side of the equation $(4x-9=0)$ yields $(4x=9)$. Dividing each side of this equation by $4$ yields $\\bm{x}=\\frac{9}{4}$. Therefore, $\\frac{9}{4}$ is a positive solution to the given equation. Subtracting $4$ from each side of the equation $\\bm{x}+\\bm{4}=0$ yields $\\bm{x}=-4$. Therefore, the given equation has two solutions, $\\frac{9}{4}$ and $-4$. 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 $\\bm{x}$ in the given equation yields $(-\\frac{49}{2}=-36)$, which is false.

Choice C is incorrect. Substituting $4$ for $\\bm{x}$ in the given equation yields $(-92=-36)$, which is false.

Choice D is incorrect. Substituting $7$ for $\\bm{x}$ in the given equation yields $(-245=-36)$, which is false.

"},"encodingFormat":"text/markdown","text":"

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

What is the positive solution to the given equation?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$\\frac{7}{4}$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$4$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$7$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$\\frac{9}{4}$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-57"}

$-4x^{2}-7x=-36$What is the positive solution to the given equation?

$-4x^{2}-7x=-36$What is the positive solution to the given equation?

$-4x^{2}-7x=-36$What is the positive solution to the given equation?

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

What is the positive solution to the given equation?

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

What is the positive solution to the given equation?

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

What is the positive solution to the given equation?