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$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

","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-55"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

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

Choice A is correct. The number of solutions to any quadratic equation in the form $ax^{2}+bx+c=0$, where a, b, and c are constants, can be found by evaluating the expression $b^{2}-4\\,ac$, which is called the discriminant. If the value of $b^{2}-4\\,ac$ is a positive number, then there will be exactly two real solutions to the equation. If the value of $b^{2}-4\\,ac$ is zero, then there will be exactly one real solution to the equation. Finally, if the value of $b^{2}-4\\,ac$ is negative, then there will be no real solutions to the equation.

The given equation $2x^{2}-4x=t$ is a quadratic equation in one variable, where $t$ is a constant. Subtracting $t$ from both sides of the equation gives $2x^{2}-4x-t=0$. In this form, $a=2,b=-4$, and $c=-t$. The values of $t$ for which the equation has no real solutions are the same values of $t$ for which the discriminant of this equation is a negative value. The discriminant is equal to $ (-4)^{2}-4(2)(-t) $, therefore, $ (-4)^{2}-4(2)(-t)<0 $. Simplifying the left side of the inequality gives $16+8t<0$. Subtracting $16$ from both sides of the inequality and then dividing both sides by $8$ gives $t<-2$. Of the values given in the options, $-3$ is the only value that is less than $-2$. Therefore, choice A must be the correct answer.

Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.

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$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

","suggestedAnswer":[{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$-1$

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

$1$

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

$3$

"}],"acceptedAnswer":{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$-3$

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

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

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-55"}

$2x^{2}-4x=t$In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

$2x^{2}-4x=t$In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

$2x^{2}-4x=t$In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?

$2x^{2}-4x=t$

In the equation above, $t$ is a constant. If the equation has no real solutions, which of the following could be the value of $t$ ?