## Finding a Second Degree Equation from Its Solutions

*This post about finding a second degree equation from its solutions is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.*

## Question

*x *= 4 is the only solution to one of the equations below.

Which one?

A.

B.

C.

D.

## Solution

### Note

For solving equations, the terms “*root*,” “*solution*,” “*zero*,” and “*x-intercept*” all mean pretty much the same thing, an answer to the problem. “*Zero*” and “*x-intercept*” come from the way equations are usually graphed for a problem: The equation is set equal to zero and then solved for the *x*-value that makes it all equal zero. On a graph, that *x*-value is an *x*-intercept.

### Back to the Question

The *factor theorem *says that if is a solution to , then must be a factor of . For example, if is a solution, must be a factor of .

For which of the answer choices is a factor? Choices B and D. What’s the right answer?

The other clue in the question is that is the **only **solution. The factor theorem says also that if is a factor of , then ** c** must be a solution to . Note that the equation in answer choice D also contains another factor: . If is a factor, then -4 is a solution. That would make a second solution — but the question says there can be only one, so D cannot be the correct answer.

That means the correct answer is choice B.

Related: “Solving a Quadratic” and “How Can You Find a Quadratic’s Equation from Its Graph?”

The Accuplacer sample problem on which this one is based is #12 in the sample questions for the Accuplacer Advanced Algebra and Functions test.