Solve for y (y+2)^2-10(y+2)=-9

${(y + 2)}^{2} - 10 (y + 2) = - 9$

Step 1

Simplify each term.

Apply the distributive property.

${(y + 2)}^{2} - 10 y - 10 \cdot 2 = - 9$

Multiply $- 10$ by $2$ .

${(y + 2)}^{2} - 10 y - 20 = - 9$

Step 2

Add $9$ to both sides of the equation.

${(y + 2)}^{2} - 10 y - 20 + 9 = 0$

Step 3

Simplify ${(y + 2)}^{2} - 10 y - 20 + 9$ .

Simplify each term.

Rewrite ${(y + 2)}^{2}$ as $(y + 2) (y + 2)$ .

$(y + 2) (y + 2) - 10 y - 20 + 9 = 0$

Expand $(y + 2) (y + 2)$ using the FOIL Method.

Apply the distributive property.

$y (y + 2) + 2 (y + 2) - 10 y - 20 + 9 = 0$

Apply the distributive property.

$y \cdot y + y \cdot 2 + 2 (y + 2) - 10 y - 20 + 9 = 0$

Apply the distributive property.

$y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 - 10 y - 20 + 9 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$y^{2} + y \cdot 2 + 2 y + 2 \cdot 2 - 10 y - 20 + 9 = 0$

Move $2$ to the left of $y$ .

$y^{2} + 2 \cdot y + 2 y + 2 \cdot 2 - 10 y - 20 + 9 = 0$

Multiply $2$ by $2$ .

$y^{2} + 2 y + 2 y + 4 - 10 y - 20 + 9 = 0$

Add $2 y$ and $2 y$ .

$y^{2} + 4 y + 4 - 10 y - 20 + 9 = 0$

Subtract $10 y$ from $4 y$ .

$y^{2} - 6 y + 4 - 20 + 9 = 0$

Subtract $20$ from $4$ .

$y^{2} - 6 y - 16 + 9 = 0$

Add $- 16$ and $9$ .

$y^{2} - 6 y - 7 = 0$

Step 4

Factor $y^{2} - 6 y - 7$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$(y - 7) (y + 1) = 0$

Step 5

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

$y - 7 = 0$

$y + 1 = 0$

Step 6

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

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

$y - 7 = 0$

Add $7$ to both sides of the equation.

$y = 7$

Step 7

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 8

The final solution is all the values that make $(y - 7) (y + 1) = 0$ true.

$y = 7, - 1$

Solve for y (y+2)^2-10(y+2)=-9

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