Solve by Factoring -y+28+y^2=2y+2y^2

$- y + 28 + y^{2} = 2 y + 2 y^{2}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 y$ from both sides of the equation.

$- y + 28 + y^{2} - 2 y = 2 y^{2}$

Subtract $2 y^{2}$ from both sides of the equation.

$- y + 28 + y^{2} - 2 y - 2 y^{2} = 0$

Step 2

Simplify $- y + 28 + y^{2} - 2 y - 2 y^{2}$ .

Subtract $2 y$ from $- y$ .

$- 3 y + 28 + y^{2} - 2 y^{2} = 0$

Subtract $2 y^{2}$ from $y^{2}$ .

$- 3 y + 28 - y^{2} = 0$

Step 3

Factor by grouping.

Reorder terms.

$- y^{2} - 3 y + 28 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 28 = - 28$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 y$ .

$- y^{2} - 3 y + 28 = 0$

Rewrite $- 3$ as $4$ plus $- 7$

$- y^{2} + (4 - 7) y + 28 = 0$

Apply the distributive property.

$- y^{2} + 4 y - 7 y + 28 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- y^{2} + 4 y) - 7 y + 28 = 0$

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

$y (- y + 4) + 7 (- y + 4) = 0$

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

$(- y + 4) (y + 7) = 0$

Step 4

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

$- y + 4 = 0$

$y + 7 = 0$

Step 5

Set $- y + 4$ equal to $0$ and solve for $y$ .

Set $- y + 4$ equal to $0$ .

$- y + 4 = 0$

Solve $- y + 4 = 0$ for $y$ .

Subtract $4$ from both sides of the equation.

$- y = - 4$

Divide each term in $- y = - 4$ by $- 1$ and simplify.

Divide each term in $- y = - 4$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 4}{- 1}$

Divide $y$ by $1$ .

$y = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$y = 4$

Step 6

Set $y + 7$ equal to $0$ and solve for $y$ .

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

Subtract $7$ from both sides of the equation.

$y = - 7$

Step 7

The final solution is all the values that make $(- y + 4) (y + 7) = 0$ true.

$y = 4, - 7$

Solve by Factoring -y+28+y^2=2y+2y^2

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