Solve for x 5-x = square root of x+7

$5 - x = \sqrt{x + 7}$

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.

$\sqrt{x + 7} = 5 - x$

Step 2

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

${\sqrt{x + 7}}^{2} = {(5 - x)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(5 - x)}^{2}$

Simplify the left side.

Simplify ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(5 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(5 - x)}^{2}$

Rewrite the expression.

${(x + 7)}^{1} = {(5 - x)}^{2}$

Simplify.

$x + 7 = {(5 - x)}^{2}$

Simplify the right side.

Simplify ${(5 - x)}^{2}$ .

Rewrite ${(5 - x)}^{2}$ as $(5 - x) (5 - x)$ .

$x + 7 = (5 - x) (5 - x)$

Expand $(5 - x) (5 - x)$ using the FOIL Method.

Apply the distributive property.

$x + 7 = 5 (5 - x) - x (5 - x)$

Apply the distributive property.

$x + 7 = 5 \cdot 5 + 5 (- x) - x (5 - x)$

Apply the distributive property.

$x + 7 = 5 \cdot 5 + 5 (- x) - x \cdot 5 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$x + 7 = 25 + 5 (- x) - x \cdot 5 - x (- x)$

Multiply $- 1$ by $5$ .

$x + 7 = 25 - 5 x - x \cdot 5 - x (- x)$

Multiply $5$ by $- 1$ .

$x + 7 = 25 - 5 x - 5 x - x (- x)$

Rewrite using the commutative property of multiplication.

$x + 7 = 25 - 5 x - 5 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$x + 7 = 25 - 5 x - 5 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$x + 7 = 25 - 5 x - 5 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$x + 7 = 25 - 5 x - 5 x + 1 x^{2}$

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

$x + 7 = 25 - 5 x - 5 x + x^{2}$

Subtract $5 x$ from $- 5 x$ .

$x + 7 = 25 - 10 x + x^{2}$

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.

$25 - 10 x + x^{2} = x + 7$

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

Subtract $x$ from both sides of the equation.

$25 - 10 x + x^{2} - x = 7$

Subtract $x$ from $- 10 x$ .

$25 + x^{2} - 11 x = 7$

Subtract $7$ from both sides of the equation.

$25 + x^{2} - 11 x - 7 = 0$

Subtract $7$ from $25$ .

$x^{2} - 11 x + 18 = 0$

Factor $x^{2} - 11 x + 18$ 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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

$(x - 9) (x - 2) = 0$

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

$x - 9 = 0$

$x - 2 = 0$

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

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

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 9) (x - 2) = 0$ true.

$x = 9, 2$

Step 5

Exclude the solutions that do not make $5 - x = \sqrt{x + 7}$ true.

$x = 2$

Solve for x 5-x = square root of x+7

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