Solve by Factoring square root of 2x+17=x+7

$\sqrt{2 x + 17} = x + 7$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{2 x + 17} - x = 7$

Subtract $7$ from both sides of the equation.

$\sqrt{2 x + 17} - x - 7 = 0$

Step 2

Move all terms not containing $\sqrt{2 x + 17}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{2 x + 17} - 7 = x$

Add $7$ to both sides of the equation.

$\sqrt{2 x + 17} = x + 7$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 x + 17}}^{2} = {(x + 7)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 17}$ as ${(2 x + 17)}^{\frac{1}{2}}$ .

${({(2 x + 17)}^{\frac{1}{2}})}^{2} = {(x + 7)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 17)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 17)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x + 17)}^{\frac{1}{2} \cdot 2} = {(x + 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 17)}^{\frac{1}{2} \cdot 2} = {(x + 7)}^{2}$

Rewrite the expression.

${(2 x + 17)}^{1} = {(x + 7)}^{2}$

Simplify.

$2 x + 17 = {(x + 7)}^{2}$

Simplify the right side.

Simplify ${(x + 7)}^{2}$ .

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$2 x + 17 = (x + 7) (x + 7)$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$2 x + 17 = x (x + 7) + 7 (x + 7)$

Apply the distributive property.

$2 x + 17 = x \cdot x + x \cdot 7 + 7 (x + 7)$

Apply the distributive property.

$2 x + 17 = x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x + 17 = x^{2} + x \cdot 7 + 7 x + 7 \cdot 7$

Move $7$ to the left of $x$ .

$2 x + 17 = x^{2} + 7 \cdot x + 7 x + 7 \cdot 7$

Multiply $7$ by $7$ .

$2 x + 17 = x^{2} + 7 x + 7 x + 49$

Add $7 x$ and $7 x$ .

$2 x + 17 = x^{2} + 14 x + 49$

Step 5

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 14 x + 49 = 2 x + 17$

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

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

$x^{2} + 14 x + 49 - 2 x = 17$

Subtract $2 x$ from $14 x$ .

$x^{2} + 12 x + 49 = 17$

Subtract $17$ from both sides of the equation.

$x^{2} + 12 x + 49 - 17 = 0$

Subtract $17$ from $49$ .

$x^{2} + 12 x + 32 = 0$

Factor $x^{2} + 12 x + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Write the factored form using these integers.

$(x + 4) (x + 8) = 0$

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

$x + 4 = 0$

$x + 8 = 0$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x + 4) (x + 8) = 0$ true.

$x = - 4, - 8$

Step 6

Exclude the solutions that do not make $\sqrt{2 x + 17} - x - 7 = 0$ true.

$x = - 4$

Solve by Factoring square root of 2x+17=x+7

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