Question

if (a,12) is a point on the graph of $y = x^2 + x$, what is a?
a = \boxed{} (use a comma to separate answers as needed.)

Explanation:

Step1: Substitute point into equation

Substitute $x=a$, $y=12$ into $y=x^2+x$:
$$12 = a^2 + a$$

Step2: Rearrange to standard quadratic form

Rewrite into $ax^2+bx+c=0$:
$$a^2 + a - 12 = 0$$

Step3: Factor the quadratic equation

Find two factors of -12 that sum to 1:
$$(a + 4)(a - 3) = 0$$

Step4: Solve for $a$

Set each factor equal to 0:
$a + 4 = 0 \implies a = -4$
$a - 3 = 0 \implies a = 3$

Answer:

3, -4