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.