Question

which constant term would mean that the expression is completely factored?
$x^{2}-3x+
-10
0
10$

Explanation:

Step1: Recall factoring quadratic form

For $ax^{2}+bx + c$ ($a = 1$ here), we look at factors of $c$ that sum to $b$.

Step2: Analyze - 10

If $c=-10$, factors of - 10 are $(-5,2)$ or $(5, - 2)$ and $-5 + 2=-3$ or $5+( - 2)=3
eq - 3$, so not completely factored.

Step3: Analyze 0

If $c = 0$, then $x^{2}-3x+0=x(x - 3)$, not in completely - factored form for non - zero constant term requirement.

Step4: Analyze 10

For $x^{2}-3x + 10$, the discriminant $\Delta=b^{2}-4ac=(-3)^{2}-4\times1\times10=9 - 40=-31<0$. So it cannot be factored over the real numbers, meaning it is completely factored over the real numbers.

Answer:

10