Solve for x 8-x = square root of 2x+19

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

Step 1

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

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

Step 2

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 3

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

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.

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

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

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

Multiply $- 1$ by $8$ .

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

Multiply $8$ by $- 1$ .

$2 x + 19 = 64 - 8 x - 8 x - x (- x)$

Rewrite using the commutative property of multiplication.

$2 x + 19 = 64 - 8 x - 8 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$2 x + 19 = 64 - 8 x - 8 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$2 x + 19 = 64 - 8 x - 8 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$2 x + 19 = 64 - 8 x - 8 x + 1 x^{2}$

Multiply $x^{2}$ by $1$ .

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

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

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

Step 4

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 5

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

$x = 3$

Solve for x 8-x = square root of 2x+19

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