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.  $\fn_phv {\color{DarkRed} \left ( 4x \right )^2=0}$

B.  $\fn_phv {\color{DarkRed} \left ( x-4 \right )^2=0}$

C.  $\fn_phv {\color{DarkRed} \left ( x+4 \right )^2=0}$

D.  $\fn_phv {\color{DarkRed} \left ( x-4 \right )\left (x+4 \right )=0}$

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 $\inline \fn_jvn x=c$  is a solution to $\inline \fn_jvn f(x)=0$, then $\inline \fn_jvn x-c$ must be a factor of $\inline \fn_jvn f(x)$. For example, if $\inline \fn_jvn x=4$ is a solution, $\inline \fn_jvn x-4$ must be a factor of $\inline \fn_jvn f(x)$.

For which of the answer choices is $\inline \fn_jvn x-4$ a factor? Choices B and D. What’s the right answer?

The other clue in the question is that $\inline \fn_jvn x=4$ is the only solution. The factor theorem says also that if $\inline \fn_jvn x-c$ is a factor of $\inline \fn_jvn f(x)$, then c must be a solution to $\inline \fn_jvn f(x)$. Note that the equation in answer choice D also contains another factor: $\inline \fn_jvn x+4$. If $\inline \fn_jvn x+4$  is a factor, then -4 is a solution. That would make $\inline \fn_jvn x=-4$ 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.

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