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

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

Step 1

Simplify each term.

Apply the distributive property.

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

Multiply $- 10$ by $2$ .

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

Step 2

Add $9$ to both sides of the equation.

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

Step 3

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

Subtract $10 x$ from $4 x$ .

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

Subtract $20$ from $4$ .

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

Add $- 16$ and $9$ .

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

Step 4

Factor $x^{2} - 6 x - 7$ 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.

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

Step 5

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

$x - 7 = 0$

$x + 1 = 0$

Step 6

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 7

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 8

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

$x = 7, - 1$

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

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