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McGraw Hill Integrated II, 2012

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McGraw Hill Integrated II, 2012 View details

10. Roots and Zeros

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Exercise 3 Page 77

Apply the difference of two squares.

Roots: - 32, 32, - 32i, 32i
Number and Type of Roots: two real roots and two imaginary roots

Practice makes perfect

We want to solve the given equation. To do so, we will apply a factoring technique called the difference of two squares. a^2-b^2 ⇔ (a+b)(a-b) To do so, we will start by writing both terms as perfect squares.

16x^4-81=0

SplitIntoFactors

Split into factors

16* x^(2*2)-81=0

a^(m* n)=(a^m)^n

16* (x^2)^2-81=0

Write as a power

4^2* (x^2)^2-9^2=0

a^m* b^m=(a * b)^m

( 4x^2)^2-9^2=0

a^2-b^2=(a+b)(a-b)

( 4x^2 + 9) ( 4x^2 - 9 ) =0

We have written the left-hand side of the equation as the product of two factors. To solve the equation, we will apply the Zero Product Property.

( 4x^2 + 9) ( 4x^2 - 9 ) =0

Use the Zero Product Property

lc4x^2+9=0 & (I) 4x^2-9=0 & (II)

▼

Solve for x

(I): LHS-9=RHS-9

l4x^2=-9 4x^2-9=0

(II): LHS+9=RHS+9

l4x^2=-9 4x^2=9

(I), (II): .LHS /4.=.RHS /4.

x^2&= -94 x^2&= 94

MoveNegNumToFrac

(I): Put minus sign in front of fraction

x^2&=- 94 x^2&= 94

(I), (II): sqrt(LHS)=sqrt(RHS)

x&=± sqrt(- 94) x&=± sqrt(94)

(I): sqrt(- a)= sqrt(a)* i

x&=± sqrt(94)* i x&=± sqrt(94)

(I), (II): sqrt(a/b)=sqrt(a)/sqrt(b)

x&=± sqrt(9)sqrt(4)* i x&=± sqrt(9)sqrt(4)

(I), (II): Calculate root

x&=± 32* i x&=± 32

Multiply

x&=± 32i x&=± 32

We found four roots of the given equation. Roots 3/2i, -3/2i, 3/2, -3/2 Two of them are real and two of them are imaginary.

Alternative Solution

Alternative Solution

We can also solve this equation by letting a=x^2. Then we can rewrite the given equation in terms of a. 16x^4-81=0 ⇔ 16a^2-81=0 Let's solve the new equation for a.

16a^2-81=0

LHS+81=RHS+81

16a^2=81

.LHS /16.=.RHS /16.

a^2 = 81/16

sqrt(LHS)=sqrt(RHS)

a=± sqrt(81/16)

sqrt(a/b)=sqrt(a)/sqrt(b)

a=± sqrt(81)/sqrt(16)

Calculate root

a=± 9/4

We found two roots for a, 94 and - 94. This means that x^2= 94 or x^2=- 94. Let's solve these two equations for x.

lcx^2=9/4 & (I) x^2=-9/4 & (II)

MoveNegFracToNum

(II): Put minus sign in numerator

x^2&= 94 x^2&= -94

MoveNegNumToFrac

(II): Put minus sign in front of fraction

x^2&= 94 x^2&=- 94

(I), (II): sqrt(LHS)=sqrt(RHS)

x&=± sqrt(94) x&=± sqrt(- 94)

(II): sqrt(- a)= sqrt(a)* i

x&=± sqrt(94) x&=± sqrt(94)* i

(I), (II): sqrt(a/b)=sqrt(a)/sqrt(b)

x&=± sqrt(9)sqrt(4) x&=± sqrt(9)sqrt(4)* i

(I), (II): Calculate root

x&=± 32 x&=± 32* i

(II): Multiply

x&=± 32 x&=± 32i

We found exactly the same four roots as before, x=± 32 and x=± 32i.

Subchapter links

Exercises p.77-79