Question

which graph represents the solutions to this equation?
$x^2 + 8x = -20$
a. graph of a parabola
b. graph of a parabola

Explanation:

Step1: Rewrite equation in standard form

Rearrange to $y = x^2 + 8x + 20$ by adding 20 to both sides.

Step2: Find vertex of the parabola

Use vertex formula $x = -\frac{b}{2a}$ for $ax^2+bx+c$. Here $a=1$, $b=8$, so:
$x = -\frac{8}{2(1)} = -4$

Step3: Calculate vertex y-value

Substitute $x=-4$ into $y = x^2 + 8x + 20$:
$y = (-4)^2 + 8(-4) + 20 = 16 - 32 + 20 = 4$
Vertex is $(-4, 4)$.

Step4: Check discriminant for x-intercepts

Discriminant: $b^2-4ac = 8^2 - 4(1)(20) = 64 - 80 = -16$
A negative discriminant means no real x-intercepts.

Answer:

B. <The parabola with vertex at (-4, 4) and no x-intercepts>