Question

factor, find the coordinates of the vertex of the related function, then graph. see example 4 32. $x^2 - 2x - 63 = 0$ 33. $x^2 + 16x + 63 = 0$

Explanation:

For problem 32: $x^2 - 2x - 63 = 0$

Step1: Factor the quadratic

Find two numbers that multiply to $-63$ and add to $-2$: $-9$ and $7$.
$(x - 9)(x + 7) = 0$

Step2: Find vertex x-coordinate

Use vertex formula $x = -\frac{b}{2a}$ for $ax^2+bx+c=0$.
$x = -\frac{-2}{2(1)} = 1$

Step3: Find vertex y-coordinate

Substitute $x=1$ into $f(x)=x^2-2x-63$.
$f(1) = (1)^2 - 2(1) - 63 = 1 - 2 - 63 = -64$

Step1: Factor the quadratic

Find two numbers that multiply to $63$ and add to $16$: $7$ and $9$.
$(x + 7)(x + 9) = 0$

Step2: Find vertex x-coordinate

Use vertex formula $x = -\frac{b}{2a}$ for $ax^2+bx+c=0$.
$x = -\frac{16}{2(1)} = -8$

Step3: Find vertex y-coordinate

Substitute $x=-8$ into $f(x)=x^2+16x+63$.
$f(-8) = (-8)^2 + 16(-8) + 63 = 64 - 128 + 63 = -1$

Answer:

Factored form: $(x - 9)(x + 7) = 0$
Vertex coordinates: $(1, -64)$
(Graph: Parabola opening upward, crossing x-axis at $(9,0)$ and $(-7,0)$, with vertex at $(1,-64)$)

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For problem 33: $x^2 + 16x + 63 = 0$