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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

2. Solving Quadratic Equations by Graphing

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Exercise 33 Page 108

Rewrite the equation leaving all the terms on one side. Factor out the greatest common factor if it exists.

Number of x-intercepts: 2
x-intercept: - 4, - 8

Practice makes perfect

We want to find how many times times the graph of the given quadratic function intersects the x-axis and what are its zeros. To do that, we will substitute 0 for y and solve the resulting equation by factoring.

Factoring

Let's start by substituting y=0.

y=x^2+12x+32

y= 0

0=x^2+12x+32

Rearrange equation

x^2+12x+32=0

▼

Factor

Write as a sum

x^2+4x+8x+32=0

Factor out x

x(x+4)+8x+32=0

Factor out 8

x(x+4)+8(x+4)=0

Factor out (x+4)

(x+8)(x+4)=0

Solving

To solve this equation, we will use the Zero Product Property.

(x+8)(x+4)=0

Use the Zero Product Property

lcx+8=0 & (I) x+4=0 & (II)

(I): LHS-8=RHS-8

lx=- 8 x+4=0

(II): LHS-4=RHS-4

lx_1=- 8 x_2=- 4

We found two solutions to the equation, which are x_1=- 8 and x_2=- 4. Therefore, the graph of y=x^2+12x+32 intersects the x-axis twice, at x=- 8 and x=- 4.

Checking Our Answer

Checking our answer

We can substitute our solutions back into the equation and simplify to check if our answers are correct. We will start with x=- 4.

x^2+12x+32=0

x= - 4

( - 4)^2+12( - 4)+32? =0

▼

Evaluate left-hand side

NegBaseToPosPow

(- a)^2 = a^2

16+12(- 4)+32? =0

a(- b)=- a * b

16+(- 48)+32? =0

a+(- b)=a-b

16-48+32? =0

Add and subtract terms

0=0 ✓

Substituting and simplifying created a true statement, so we know that x=- 4 is a solution of the equation. Let's move on to x=- 8.

x^2+12x+32=0

x= - 8

( - 8)^2+12( - 8)+32? =0

▼

Evaluate left-hand side

NegBaseToPosPow

(- a)^2 = a^2

64+12(- 8)+32? =0

a(- b)=- a * b

64+(- 96)+32? =0

a+(- b)=a-b

64-96+32? =0

Add and subtract terms

0=0 ✓

Again, we created a true statement. This means that x=- 8 is indeed a solution of the equation.

Subchapter links

Exercises p.108-110