{"@context":"https://schema.org/","@type":"Quiz","name":"

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

","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-7"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

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

Choice B is correct. Adding 9 to each side of the given equation yields $x^{2}-2x=9$. To complete the square, adding 1 to each side of this equation yields $x^{2}-2x+1=9+1$, or $\\left(x-1\\right)^{2}=10$. Taking the square root of each side of this equation yields $x-1=\\pm\\sqrt{10}$. Adding 1 to each side of this equation yields $x=1\\pm\\sqrt{10}$. Since it's given that one of the solutions to the equation can be written as $1+\\sqrt{k}$, the value of $k$ must be 10.

Alternate approach: It's given that $1+\\sqrt{k}$ is a solution to the given equation. It follows that $x=1+\\sqrt{k}$.Substituting $1+\\sqrt{k}$ for $x$ in the given equation yields $(1+\\sqrt{k})^{2}-2(1+\\sqrt{k})-9=0$, or $(1+\\sqrt{k})(1+\\sqrt{k})-2(1+\\sqrt{k})-9=0$. Expanding the products on the left-hand side of this equation yields $1+2\\sqrt{k}+k-2-2\\sqrt{k}-9=0$, or $k-10=0$. Adding $10$ to each side of this equation yields $k=10$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

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

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

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

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

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

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

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

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

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

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-7"}

$x^{2}-2x-9=0$One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$One solution to the given equation can be written as $1+\sqrt{k}$, where $k$ is a constant. What is the value of $k$?

$x^{2}-2x-9=0$

One solution to the given equation can be written as $1+\sqrt{k}$, where $k$ is a constant. What is the value of $k$?