Exercise 34 Page 637

To solve equations with a variable expression inside a radical, we first want to make sure the radical is isolated. Then we can raise both sides of the equation to a power equal to the index of the radical. Let's try to solve our equation using this method! We now have a quadratic equation, and we need to find its roots. To do it, let's identify the values of a, b, and c. 4y^2-5y-6=0 ⇕ 4y^2+( - 5)y+( -6)=0We can see that a= 4, b= - 5, and c= -6. Let's substitute these values into the Quadratic Formula.

y=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

y=- ( -5)±sqrt(( - 5)^2-4( 4)( -6))/2( 4)

▼

Evaluate

NegNeg

- (- a)=a

y=5±sqrt((- 5)^2-4(4)(-6))/2(4)

NegBaseToPosBase

(- a)^2=a^2

y=5±sqrt(25-4(4)(-6))/2(4)

Multiply

Multiply

y=5±sqrt(25-16(-6))/8

MultNegNegOnePar

- a(- b)=a* b

y=5±sqrt(25+96)/8

AddTerms

Add terms

y=5±sqrt(121)/8

CalcRoot

Calculate root

y=5± 11/8

Using the Quadratic Formula, we found that the solutions of the given equation are y= 5± 11 8.

y=5± 11/8
y_1=5+11/8	y_2=5-11/8
y_1=16/8	y_2=-6/8
y_1= 2	y_2= -3/4

Therefore, the solutions are y_1=2 and y_2=- 34. Let's check them to see if we have any extraneous solutions.

Checking the Solutions

We will check y_1=2 and y_2=- 34 one at a time.

y_1=2

Let's substitute y=2 into the original equation.

2y=sqrt(5y+6)

Substitute

y= 2

2( 2)? =sqrt(5( 2)+6)

▼

Simplify

Multiply

Multiply

4? =sqrt(10+6)

AddTerms

Add terms

4? =sqrt(16)

CalcRoot

Calculate root

4=4 ✓

In this case we got a true statement. Therefore, y=2 is a solution of the original equation.

y_2=- 34

Now, let's substitute y= - 34.

2y=sqrt(5y+6)

Substitute

y= -3/4

2( -3/4)? =sqrt(5( -3/4)+6)

▼

Simplify

MultPosNeg

a(- b)=- a * b

-2 * 3/4? =sqrt(- 5* 3/4+6)

MoveLeftFacToNum

a*b/c= a* b/c

- 6/4? =sqrt(- 15/4+6)

NumberToFrac

a = 4* a/4

- 6/4? =sqrt(- 15/4+4*6/4)

Multiply

Multiply

- 6/4? =sqrt(- 15/4+24/4)

CommutativePropAdd

Commutative Property of Addition

- 6/4? =sqrt(24/4 - 15/4)

SubFrac

Subtract fractions

- 6/4? =sqrt(9/4)

SqrtQuot

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

- 6/4? =sqrt(9)/sqrt(4)

CalcRoot

Calculate root

-6/4? =3/2

ReduceFrac

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

-3/2 ≠ 3/2 *

We got a false statement, so y=- 34 is an extraneous solution. Therefore, y=2 is the only solution of the original equation.