Exercise 37 Page 696

We want to solve the given rational equation. s/3s+2+s+3/2s-4=-2 s/3s^2-4s-4 We will start by factoring the denominators to find the least common denominator (LCD). Note that the first denominator is already factored. Let's factor the second one.

2s-4

FactorOut

Factor out 2

2(s-2)

Now let's factor the third denominator.

3s^2-4s-4

WriteSum

Write as a sum

3s^2-6s+2s-4

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Factor out 3s & 2

FactorOut

Factor out 3s

3s(s-2)+2s-4

FactorOut

Factor out 2

3s(s-2)+2(s-2)

FactorOut

Factor out (s-2)

(s-2)(3s+2)

We can now rewrite the given expression. s/3s+2+s+3/2s-4=-2 s/3s^2-4s-4 ⇕ s/3s+2+s+3/2 (s-2)=-2 s/(s-2) (3s+2) We will solve the equation by multiplying each side by the LCD to clear denominators.

s/3s+2+s+3/2(s-2)=-2 s/(s-2)(3s+2)

MultEqn

LHS * 2 (3s+2) (s-2)=RHS* 2 (3s+2) (s-2)

(s/3s+2+s+3/2(s-2)) * 2 (3s+2) (s-2)=-2 s/(s-2)(3s+2) * 2 (3s+2) (s-2)

Distr

Distribute 2(3s+2)(s-2)

s/3s+2 * 2(3s+2)(s-2)+s+3/2(s-2)* 2(3s+2)(s-2)=-2 s/(s-2)(3s+2)* 2(3s+2)(s-2)

MoveRightFacToNum

a/c* b = a* b/c

s(2)(3s+2)(s-2)/3s+2 +(s+3)(2)(3s+2)(s-2)/2(s-2)=-2 s(2)(3s+2)(s-2)/(s-2)(3s+2)

CancelCommonFac

Cancel out common factors

s(2)(3s+2)(s-2)/3s+2 +(s+3)(2)(3s+2)(s-2)/2(s-2)=-2 s(2)(3s+2)(s-2)/(s-2)(3s+2)

SimpQuot

Simplify quotient

s(2)(s-2)+(s+3)(3s+2)=-2 s(2)

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Simplify

CommutativePropMult

Commutative Property of Multiplication

2s(s-2)+(s+3)(3s+2)=-4 s

Distr

Distribute (3s+2)

2s(s-2)+s(3s+2)+3(3s+2)=-4 s

Distr

Distribute 2s

2s^2-4s+s(3s+2)+3(3s+2)=-4 s

Distr

Distribute s

2s^2-4s+3s^2+2s+3(3s+2)=-4 s

Distr

Distribute 3

2s^2-4s+3s^2+2s+9s+6=-4 s

AddEqn

LHS+4s=RHS+4s

2s^2+3s^2+2s+9s+6=0

AddTerms

Add terms

5s^2+11s+6=0

We obtained a quadratic equation. Let's identify the values of a, b, and c. 5s^2+ 11s+ 6=0 We see that a = 5, b = 11, and c = 6. Next, we will substitute these values into the Quadratic Formula.

s=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

s=- 11±sqrt(11^2-4( 5)( 6))/2( 5)

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Solve using the quadratic formula

CalcPow

Calculate power

s=-11±sqrt(121-4(5)(6))/2(5)

Multiply

s=-11±sqrt(121-120)/10

SubTerm

Subtract term

s=-11±sqrt(1)/10

CalcRoot

Calculate root

s=-11± 1/10

We will find the values for s by using the positive and the negative signs.

s=-11± 1/10
s=-11+ 1/10	s=-11- 1/10
s=-10/10	s=-12/10
s=-1	s=-6/5

We found that s=-1 and s=- 65 are possible solutions. Now we have to check them!

Checking the Solutions

We check our solutions to see whether any of them are extraneous. To do so, we will substitute s=-1 and s=- 65 into the given equation. Let's start with s=-1.

s/3s+2+s+3/2s-4=-2 s/3s^2-4s-4

Substitute

s= -1

-1/3( -1)+2+-1+3/2( -1)-4? =-2 ( -1)/3( -1)^2-4( -1)-4

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Simplify

NegBaseToPosBase

(- a)^2=a^2

-1/3(-1)+2+-1+3/2(-1)-4? =-2 (-1)/3(1)-4(-1)-4

IdPropMult

Identity Property of Multiplication

-1/3(-1)+2+-1+3/2(-1)-4? =-2 (-1)/3-4(-1)-4

MultPosNeg

a(- b)=- a * b

-1/-3+2+-1+3/-2-4? =-2 (-1)/3-4(-1)-4

MultNegNegOnePar

- a(- b)=a* b

-1/-3+2+-1+3/-2-4? =2/3+4-4

AddSubTerms

Add and subtract terms

-1/-1+2/-6? =2/3

DivNegNeg

- a/- b=a/b

1/1+2/-6? =2/3

MoveNegDenomToFrac

Put minus sign in front of fraction

1/1+(-2/6)? =2/3

AddNeg

a+(- b)=a-b

1/1-2/6? =2/3

ReduceFrac

a/b=.a /2./.b /2.

1/1-1/3? =2/3

ExpandFrac

a/b=a * 3/b * 3

3/3-1/3? =2/3

SubFrac

Subtract fractions

2/3=2/3 ✓

Since we obtained a true statement, s=-1 is a solution of the equation. Let's now substitute s=- 65 into the equation.

s/3s+2+s+3/2s-4=-2 s/3s^2-4s-4

Substitute

s= -6/5

- 65/3( - 65)+2+- 65+3/2( - 65)-4? =-2 ( - 65)/3( - 65)^2-4( - 65)-4

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Simplify

NegBaseToPosBase

(- a)^2=a^2

- 65/3(- 65)+2+- 65+3/2(- 65)-4? =-2 (- 65)/3( 65)^2-4(- 65)-4

PowQuot

(a/b)^m=a^m/b^m

- 65/3(- 65)+2+- 65+3/2(- 65)-4? =-2 (- 65)/3( 3625)-4(- 65)-4

MultPosNeg

a(- b)=- a * b

- 65/-3( 65)+2+- 65+3/-2( 65)-4? =-2(- 65)/3( 3625)-4(- 65)-4

MultNegNegOnePar

- a(- b)=a* b

- 65/-3( 65)+2+- 65+3/-2( 65)-4? =2 ( 65)/3( 3625)+4( 65)-4

MoveLeftFacToNum

a*b/c= a* b/c

- 65/- 185+2+- 65+3/- 125-4? =125/10825+ 245-4

NumberToFrac

a = 5* a/5

- 65/- 185+ 105+- 65+ 155/- 125- 205? =125/10825+ 245- 205

ExpandFrac

a/b=a * 5/b * 5

- 65/- 185+ 105+- 65+ 155/- 125- 205? =125/10825+ 12025- 10025

MoveNegFracToNum

Put minus sign in numerator

-65/-185+ 105+-65+ 155/-125- 205? =125/10825+ 12025- 10025

AddSubTerms

Add and subtract terms

-65/-85+95/-325? =125/12825

MoveNegNumToFrac

Put minus sign in front of fraction

- 65/- 85+95/- 325? =125/12825

DivNegNeg

- a/- b=a/b

65/85+95/- 325? =125/12825

Recall that dividing by a fraction is the same as multiplying by its reciprocal. 65/85+95/- 325? =125/12825 ⇕ 6/5(5/8)+9/5(-5/32)? = 12/5(25/128) Let's simplify to finally see whether the equation is satisfied when s=- 65.

6/5(5/8)+9/5(-5/32)? = 12/5(25/128)

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Simplify

MultPosNeg

a(- b)=- a * b

6/5(5/8)-9/5(5/32)? = 12/5(25/128)

MultFrac

Multiply fractions

30/40-45/160? = 300/640

ReduceFrac

a/b=.a /10./.b /10.

3/4-45/160? = 300/640