Exercise 78 Page 393

a^2-b^2 ⇔ (a+b)(a-b) To do so, we will start by simplifying the equation to obtain the difference of two terms. Then, we will write both terms as perfect squares.

4+3x^4=81

SubEqn

LHS-4=RHS-4

3x^4 = 77

DivEqn

.LHS /3.=.RHS /3.

x^4 = 77/3

SubEqn

LHS-77/3=RHS-77/3

x^4 - 77/3 = 0

SplitIntoFactors

Split into factors

x^(2*2) - 77/3 = 0

ProdInExponent

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

( x^2 ) ^2-77/3=0

a = ( sqrt(a) )^2

( x^2 ) ^2-(sqrt(77/3))^2=0

FacDiffSquares

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

( x^2+sqrt(77/3) ) ( x^2-sqrt(77/3) ) =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.

( x^2+sqrt(77/3) ) ( x^2-sqrt(77/3) ) =0

ZeroProdProp

Use the Zero Product Property

lcx^2+sqrt(77/3)=0 & (I) x^2-sqrt(77/3)=0 & (II)

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(I), (II): Solve for x

SubEqn

(I): LHS-sqrt(77/3)=RHS-sqrt(77/3)

lx^2=-sqrt(77/3) x^2-sqrt(77/3)=0

AddEqn

(II): LHS+sqrt(77/3)=RHS+sqrt(77/3)

lx^2=-sqrt(77/3) x^2=sqrt(77/3)

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

lx=±sqrt(-sqrt(77/3)) x=±sqrt(sqrt(77/3))

SqrtNegToISqrt

(I): sqrt(- a)= isqrt(a)

lx=± isqrt(sqrt(77/3)) x=±sqrt(sqrt(77/3))

(I), (II): sqrt(a)=a^(12)

lx=± i((77/3)^(12))^(12) x=±((77/3)^(12))^(12)

(I), (II): (a^m)^n=a^(m* n)

lx=± i(77/3)^(12* 12) x=±(77/3)^(12* 12)

(I), (II):Multiply fractions

lx=± i(77/3)^(14) x=±(77/3)^(14)

(I), (II): a^(1n)=sqrt(a)

lx=± isqrt(77/3) x=±sqrt(77/3)

CommutativePropMult

(I): Commutative Property of Multiplication

lx=±sqrt(77/3)i x=±sqrt(77/3)

(I), (II): Use a calculator

lx≈± 2.25i x≈± 2.25

We found four approximate solutions for the given equation. 2.25, - 2.25, 2.25i, - 2.25i