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

Math
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
Simplify each side of the equation.
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Multiply the exponents in ((7y2+27y)12)2.
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Apply the power rule and multiply exponents, (am)n=amn.
(7y2+27y)12⋅2=3y2+402
Cancel the common factor of 2.
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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.
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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.
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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
Solve for y.
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Move all terms containing y to the left side of the equation.
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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.
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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.
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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.
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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.
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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.
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Divide each term in 4y=5 by 4.
4y4=54
Cancel the common factor of 4.
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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.
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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

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