7y2+27y=3y2+40

To remove the radical on the left side of the equation, square both sides of the equation.

7y2+27y2=3y2+402

Multiply the exponents in ((7y2+27y)12)2.

Apply the power rule and multiply exponents, (am)n=amn.

(7y2+27y)12⋅2=3y2+402

Cancel the common factor of 2.

Cancel the common factor.

(7y2+27y)12⋅2=3y2+402

Rewrite the expression.

(7y2+27y)1=3y2+402

(7y2+27y)1=3y2+402

(7y2+27y)1=3y2+402

Simplify.

7y2+27y=3y2+402

Rewrite 3y2+402 as 3y2+40.

Use axn=axn to rewrite 3y2+40 as (3y2+40)12.

7y2+27y=((3y2+40)12)2

Apply the power rule and multiply exponents, (am)n=amn.

7y2+27y=(3y2+40)12⋅2

Combine 12 and 2.

7y2+27y=(3y2+40)22

Cancel the common factor of 2.

Cancel the common factor.

7y2+27y=(3y2+40)22

Divide 1 by 1.

7y2+27y=(3y2+40)1

7y2+27y=(3y2+40)1

Simplify.

7y2+27y=3y2+40

7y2+27y=3y2+40

7y2+27y=3y2+40

Move all terms containing y to the left side of the equation.

Subtract 3y2 from both sides of the equation.

7y2+27y-3y2=40

Subtract 3y2 from 7y2.

4y2+27y=40

4y2+27y=40

Move 40 to the left side of the equation by subtracting it from both sides.

4y2+27y-40=0

Factor by grouping.

For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=4⋅-40=-160 and whose sum is b=27.

Factor 27 out of 27y.

4y2+27(y)-40=0

Rewrite 27 as -5 plus 32

4y2+(-5+32)y-40=0

Apply the distributive property.

4y2-5y+32y-40=0

4y2-5y+32y-40=0

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

(4y2-5y)+32y-40=0

Factor out the greatest common factor (GCF) from each group.

y(4y-5)+8(4y-5)=0

y(4y-5)+8(4y-5)=0

Factor the polynomial by factoring out the greatest common factor, 4y-5.

(4y-5)(y+8)=0

(4y-5)(y+8)=0

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

4y-5=0

y+8=0

Set the first factor equal to 0 and solve.

Set the first factor equal to 0.

4y-5=0

Add 5 to both sides of the equation.

4y=5

Divide each term by 4 and simplify.

Divide each term in 4y=5 by 4.

4y4=54

Cancel the common factor of 4.

Cancel the common factor.

4y4=54

Divide y by 1.

y=54

y=54

y=54

y=54

Set the next factor equal to 0 and solve.

Set the next factor equal to 0.

y+8=0

Subtract 8 from both sides of the equation.

y=-8

y=-8

The final solution is all the values that make (4y-5)(y+8)=0 true.

y=54,-8

y=54,-8

The result can be shown in multiple forms.

Exact Form:

y=54,-8

Decimal Form:

y=1.25,-8

Mixed Number Form:

y=114,-8

Solve for y square root of 7y^2+27y = square root of 3y^2+40