Solve for y -7y=2y^2+5

-7y=2y2+5
Subtract 2y2 from both sides of the equation.
-7y-2y2=5
Move 5 to the left side of the equation by subtracting it from both sides.
-7y-2y2-5=0
Factor the left side of the equation.
Let u=y. Substitute u for all occurrences of y.
-7u-2u2-5
Factor by grouping.
Reorder terms.
-2u2-7u-5
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=-2⋅-5=10 and whose sum is b=-7.
Factor -7 out of -7u.
-2u2-7(u)-5
Rewrite -7 as -2 plus -5
-2u2+(-2-5)u-5
Apply the distributive property.
-2u2-2u-5u-5
-2u2-2u-5u-5
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(-2u2-2u)-5u-5
Factor out the greatest common factor (GCF) from each group.
2u(-u-1)+5(-u-1)
2u(-u-1)+5(-u-1)
Factor the polynomial by factoring out the greatest common factor, -u-1.
(-u-1)(2u+5)
(-u-1)(2u+5)
Replace all occurrences of u with y.
(-y-1)(2y+5)
Replace the left side with the factored expression.
(-y-1)(2y+5)=0
(-y-1)(2y+5)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
-y-1=0
2y+5=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
-y-1=0
Add 1 to both sides of the equation.
-y=1
Multiply each term in -y=1 by -1
Multiply each term in -y=1 by -1.
(-y)⋅-1=1⋅-1
Multiply (-y)⋅-1.
Multiply -1 by -1.
1y=1⋅-1
Multiply y by 1.
y=1⋅-1
y=1⋅-1
Multiply -1 by 1.
y=-1
y=-1
y=-1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
2y+5=0
Subtract 5 from both sides of the equation.
2y=-5
Divide each term by 2 and simplify.
Divide each term in 2y=-5 by 2.
2y2=-52
Cancel the common factor of 2.
Cancel the common factor.
2y2=-52
Divide y by 1.
y=-52
y=-52
Move the negative in front of the fraction.
y=-52
y=-52
y=-52
The final solution is all the values that make (-y-1)(2y+5)=0 true.
y=-1,-52
The result can be shown in multiple forms.
Exact Form:
y=-1,-52
Decimal Form:
y=-1,-2.5
Mixed Number Form:
y=-1,-212
Solve for y -7y=2y^2+5

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