Solve by Factoring square root of 2x+19=x+8

$\sqrt{2 x + 19} = x + 8$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{2 x + 19} - x = 8$

Subtract $8$ from both sides of the equation.

$\sqrt{2 x + 19} - x - 8 = 0$

Step 2

Move all terms not containing $\sqrt{2 x + 19}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{2 x + 19} - 8 = x$

Add $8$ to both sides of the equation.

$\sqrt{2 x + 19} = x + 8$

Step 3

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

${\sqrt{2 x + 19}}^{2} = {(x + 8)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 19}$ as ${(2 x + 19)}^{\frac{1}{2}}$ .

${({(2 x + 19)}^{\frac{1}{2}})}^{2} = {(x + 8)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 19)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 19)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 19)}^{\frac{1}{2} \cdot 2} = {(x + 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 19)}^{\frac{1}{2} \cdot 2} = {(x + 8)}^{2}$

Rewrite the expression.

${(2 x + 19)}^{1} = {(x + 8)}^{2}$

Simplify.

$2 x + 19 = {(x + 8)}^{2}$

Simplify the right side.

Simplify ${(x + 8)}^{2}$ .

Rewrite ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$2 x + 19 = (x + 8) (x + 8)$

Expand $(x + 8) (x + 8)$ using the FOIL Method.

Apply the distributive property.

$2 x + 19 = x (x + 8) + 8 (x + 8)$

Apply the distributive property.

$2 x + 19 = x \cdot x + x \cdot 8 + 8 (x + 8)$

Apply the distributive property.

$2 x + 19 = x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x + 19 = x^{2} + x \cdot 8 + 8 x + 8 \cdot 8$

Move $8$ to the left of $x$ .

$2 x + 19 = x^{2} + 8 \cdot x + 8 x + 8 \cdot 8$

Multiply $8$ by $8$ .

$2 x + 19 = x^{2} + 8 x + 8 x + 64$

Add $8 x$ and $8 x$ .

$2 x + 19 = x^{2} + 16 x + 64$

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.

$x^{2} + 16 x + 64 = 2 x + 19$

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

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

$x^{2} + 16 x + 64 - 2 x = 19$

Subtract $2 x$ from $16 x$ .

$x^{2} + 14 x + 64 = 19$

Subtract $19$ from both sides of the equation.

$x^{2} + 14 x + 64 - 19 = 0$

Subtract $19$ from $64$ .

$x^{2} + 14 x + 45 = 0$

Factor $x^{2} + 14 x + 45$ 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 $45$ and whose sum is $14$ .

$5, 9$

Write the factored form using these integers.

$(x + 5) (x + 9) = 0$

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

$x + 5 = 0$

$x + 9 = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $(x + 5) (x + 9) = 0$ true.

$x = - 5, - 9$

Step 6

Exclude the solutions that do not make $\sqrt{2 x + 19} - x - 8 = 0$ true.

$x = - 5$

Solve by Factoring square root of 2x+19=x+8

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