Question

worksheet #1 - solving quadratics by factoring

1. $x^2 - 8x + 12$

factored form
x intercepts
y intercept
vertex

1. $-x^2 + 10x - 21$

factored form
x intercepts
y intercept
vertex

1. $x^2 - 8x + 15$

factored form
x intercepts
y intercept
vertex

1. $x^2 - x - 30$

factored form
x intercepts
y intercept
vertex

1. $x^2 - 9x - 10$

factored form
x intercepts
y intercept
vertex

Explanation:

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Problem 1: $x^2 - 8x + 12$

Step1: Factor the quadratic

Find two numbers that multiply to 12 and add to -8: -2 and -6.
$\boldsymbol{x^2 - 8x + 12 = (x-2)(x-6)}$

Step2: Find x-intercepts

Set factored form = 0, solve for $x$.
$x-2=0 \implies x=2$; $x-6=0 \implies x=6$

Step3: Find y-intercept

Set $x=0$, evaluate the quadratic.
$y=(0)^2 -8(0)+12=12$

Step4: Find vertex

Use $x=-\frac{b}{2a}$, then find $y$.
$x=-\frac{-8}{2(1)}=4$; $y=(4)^2-8(4)+12=16-32+12=-4$

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Problem 2: $-x^2 + 10x -21$

Step1: Factor out -1 first

$\boldsymbol{-x^2 +10x -21 = -(x^2 -10x +21)}$

Step2: Factor the quadratic inside

Find two numbers that multiply to 21 and add to -10: -3 and -7.
$\boldsymbol{-(x-3)(x-7)}$

Step3: Find x-intercepts

Set factored form = 0, solve for $x$.
$x-3=0 \implies x=3$; $x-7=0 \implies x=7$

Step4: Find y-intercept

Set $x=0$, evaluate the quadratic.
$y=-(0)^2+10(0)-21=-21$

Step5: Find vertex

Use $x=-\frac{b}{2a}$, then find $y$.
$x=-\frac{10}{2(-1)}=5$; $y=-(5)^2+10(5)-21=-25+50-21=4$

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Problem 3: $x^2 -8x +15$

Step1: Factor the quadratic

Find two numbers that multiply to 15 and add to -8: -3 and -5.
$\boldsymbol{x^2 -8x +15=(x-3)(x-5)}$

Step2: Find x-intercepts

Set factored form = 0, solve for $x$.
$x-3=0 \implies x=3$; $x-5=0 \implies x=5$

Step3: Find y-intercept

Set $x=0$, evaluate the quadratic.
$y=(0)^2-8(0)+15=15$

Step4: Find vertex

Use $x=-\frac{b}{2a}$, then find $y$.
$x=-\frac{-8}{2(1)}=4$; $y=(4)^2-8(4)+15=16-32+15=-1$

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Problem 4: $x^2 -x -30$

Step1: Factor the quadratic

Find two numbers that multiply to -30 and add to -1: 5 and -6.
$\boldsymbol{x^2 -x -30=(x+5)(x-6)}$

Step2: Find x-intercepts

Set factored form = 0, solve for $x$.
$x+5=0 \implies x=-5$; $x-6=0 \implies x=6$

Step3: Find y-intercept

Set $x=0$, evaluate the quadratic.
$y=(0)^2-(0)-30=-30$

Step4: Find vertex

Use $x=-\frac{b}{2a}$, then find $y$.
$x=-\frac{-1}{2(1)}=0.5$; $y=(0.5)^2-(0.5)-30=0.25-0.5-30=-30.25$

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Problem 5: $x^2 -9x -10$

Step1: Factor the quadratic

Find two numbers that multiply to -10 and add to -9: 1 and -10.
$\boldsymbol{x^2 -9x -10=(x+1)(x-10)}$

Step2: Find x-intercepts

Set factored form = 0, solve for $x$.
$x+1=0 \implies x=-1$; $x-10=0 \implies x=10$

Step3: Find y-intercept

Set $x=0$, evaluate the quadratic.
$y=(0)^2-9(0)-10=-10$

Step4: Find vertex

Use $x=-\frac{b}{2a}$, then find $y$.
$x=-\frac{-9}{2(1)}=4.5$; $y=(4.5)^2-9(4.5)-10=20.25-40.5-10=-30.25$

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Answer:

1.

• Factored Form: $(x-2)(x-6)$
• X intercepts: $x=2$, $x=6$
• Y intercept: $y=12$
• Vertex: $(4, -4)$

2.

• Factored Form: $-(x-3)(x-7)$
• X intercepts: $x=3$, $x=7$
• Y intercept: $y=-21$
• Vertex: $(5, 4)$

3.

• Factored Form: $(x-3)(x-5)$
• X intercepts: $x=3$, $x=5$
• Y intercept: $y=15$
• Vertex: $(4, -1)$

4.

• Factored Form: $(x+5)(x-6)$
• X intercepts: $x=-5$, $x=6$
• Y intercept: $y=-30$
• Vertex: $(0.5, -30.25)$

5.

• Factored Form: $(x+1)(x-10)$
• X intercepts: $x=-1$, $x=10$
• Y intercept: $y=-10$
• Vertex: $(4.5, -30.25)$