Solve for x x^2+(2x+4)^2=(2x+6)^2

$x^{2} + {(2 x + 4)}^{2} = {(2 x + 6)}^{2}$

Step 1

Simplify ${(2 x + 6)}^{2}$ .

Rewrite ${(2 x + 6)}^{2}$ as $(2 x + 6) (2 x + 6)$ .

$x^{2} + {(2 x + 4)}^{2} = (2 x + 6) (2 x + 6)$

Expand $(2 x + 6) (2 x + 6)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + {(2 x + 4)}^{2} = 2 x (2 x + 6) + 6 (2 x + 6)$

Apply the distributive property.

$x^{2} + {(2 x + 4)}^{2} = 2 x (2 x) + 2 x \cdot 6 + 6 (2 x + 6)$

Apply the distributive property.

$x^{2} + {(2 x + 4)}^{2} = 2 x (2 x) + 2 x \cdot 6 + 6 (2 x) + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} + {(2 x + 4)}^{2} = 2 \cdot 2 x \cdot x + 2 x \cdot 6 + 6 (2 x) + 6 \cdot 6$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} + {(2 x + 4)}^{2} = 2 \cdot 2 (x \cdot x) + 2 x \cdot 6 + 6 (2 x) + 6 \cdot 6$

Multiply $x$ by $x$ .

$x^{2} + {(2 x + 4)}^{2} = 2 \cdot 2 x^{2} + 2 x \cdot 6 + 6 (2 x) + 6 \cdot 6$

Multiply $2$ by $2$ .

$x^{2} + {(2 x + 4)}^{2} = 4 x^{2} + 2 x \cdot 6 + 6 (2 x) + 6 \cdot 6$

Multiply $6$ by $2$ .

$x^{2} + {(2 x + 4)}^{2} = 4 x^{2} + 12 x + 6 (2 x) + 6 \cdot 6$

Multiply $2$ by $6$ .

$x^{2} + {(2 x + 4)}^{2} = 4 x^{2} + 12 x + 12 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$x^{2} + {(2 x + 4)}^{2} = 4 x^{2} + 12 x + 12 x + 36$

Add $12 x$ and $12 x$ .

$x^{2} + {(2 x + 4)}^{2} = 4 x^{2} + 24 x + 36$

Step 2

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$x^{2} + {(2 x + 4)}^{2} - 4 x^{2} = 24 x + 36$

Subtract $24 x$ from both sides of the equation.

$x^{2} + {(2 x + 4)}^{2} - 4 x^{2} - 24 x = 36$

Simplify each term.

Rewrite ${(2 x + 4)}^{2}$ as $(2 x + 4) (2 x + 4)$ .

$x^{2} + (2 x + 4) (2 x + 4) - 4 x^{2} - 24 x = 36$

Expand $(2 x + 4) (2 x + 4)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 2 x (2 x + 4) + 4 (2 x + 4) - 4 x^{2} - 24 x = 36$

Apply the distributive property.

$x^{2} + 2 x (2 x) + 2 x \cdot 4 + 4 (2 x + 4) - 4 x^{2} - 24 x = 36$

Apply the distributive property.

$x^{2} + 2 x (2 x) + 2 x \cdot 4 + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} + 2 \cdot 2 x \cdot x + 2 x \cdot 4 + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} + 2 \cdot 2 (x \cdot x) + 2 x \cdot 4 + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $x$ by $x$ .

$x^{2} + 2 \cdot 2 x^{2} + 2 x \cdot 4 + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $2$ by $2$ .

$x^{2} + 4 x^{2} + 2 x \cdot 4 + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $4$ by $2$ .

$x^{2} + 4 x^{2} + 8 x + 4 (2 x) + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $2$ by $4$ .

$x^{2} + 4 x^{2} + 8 x + 8 x + 4 \cdot 4 - 4 x^{2} - 24 x = 36$

Multiply $4$ by $4$ .

$x^{2} + 4 x^{2} + 8 x + 8 x + 16 - 4 x^{2} - 24 x = 36$

Add $8 x$ and $8 x$ .

$x^{2} + 4 x^{2} + 16 x + 16 - 4 x^{2} - 24 x = 36$

Combine the opposite terms in $x^{2} + 4 x^{2} + 16 x + 16 - 4 x^{2} - 24 x$ .

Subtract $4 x^{2}$ from $4 x^{2}$ .

$x^{2} + 16 x + 16 + 0 - 24 x = 36$

Add $x^{2} + 16 x + 16$ and $0$ .

$x^{2} + 16 x + 16 - 24 x = 36$

Subtract $24 x$ from $16 x$ .

$x^{2} - 8 x + 16 = 36$

Step 3

Subtract $36$ from both sides of the equation.

$x^{2} - 8 x + 16 - 36 = 0$

Step 4

Subtract $36$ from $16$ .

$x^{2} - 8 x - 20 = 0$

Step 5

Factor $x^{2} - 8 x - 20$ 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 $- 20$ and whose sum is $- 8$ .

$- 10, 2$

Write the factored form using these integers.

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

Step 6

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

$x - 10 = 0$

$x + 2 = 0$

Step 7

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

Step 8

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9

The final solution is all the values that make $(x - 10) (x + 2) = 0$ true.

$x = 10, - 2$

Solve for x x^2+(2x+4)^2=(2x+6)^2

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