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

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

Step 1

Subtract $\sqrt{2 x + 19}$ from both sides of the equation.

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

Step 2

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

Subtract $8$ from both sides of the equation.

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

Add $x$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $- {(2 x + 19)}^{\frac{1}{2}}$ .

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

Raise $- 1$ to the power of $2$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 8$ by $- 8$ .

$2 x + 19 = 64 - 8 x + x \cdot - 8 + x \cdot x$

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

$2 x + 19 = 64 - 8 x - 8 \cdot x + x \cdot x$

Multiply $x$ by $x$ .

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

Subtract $8 x$ from $- 8 x$ .

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

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.

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

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

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

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

Subtract $2 x$ from $- 16 x$ .

$64 + x^{2} - 18 x = 19$

Subtract $19$ from both sides of the equation.

$64 + x^{2} - 18 x - 19 = 0$

Subtract $19$ from $64$ .

$x^{2} - 18 x + 45 = 0$

Factor $x^{2} - 18 x + 45$ 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 $- 18$ .

$- 15, - 3$

Write the factored form using these integers.

$(x - 15) (x - 3) = 0$

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

$x - 15 = 0$

$x - 3 = 0$

Set $x - 15$ equal to $0$ and solve for $x$ .

Set $x - 15$ equal to $0$ .

$x - 15 = 0$

Add $15$ to both sides of the equation.

$x = 15$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x - 15) (x - 3) = 0$ true.

$x = 15, 3$

Step 6

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

$x = 3$

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

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