Solve by Factoring square root of 7x+36=x+4

$\sqrt{7 x + 36} = x + 4$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{7 x + 36} - x = 4$

Subtract $4$ from both sides of the equation.

$\sqrt{7 x + 36} - x - 4 = 0$

Step 2

Move all terms not containing $\sqrt{7 x + 36}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{7 x + 36} - 4 = x$

Add $4$ to both sides of the equation.

$\sqrt{7 x + 36} = x + 4$

Step 3

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

${\sqrt{7 x + 36}}^{2} = {(x + 4)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x + 36}$ as ${(7 x + 36)}^{\frac{1}{2}}$ .

${({(7 x + 36)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Simplify the left side.

Simplify ${({(7 x + 36)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(7 x + 36)}^{\frac{1}{2}})}^{2}$ .

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

${(7 x + 36)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 x + 36)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Rewrite the expression.

${(7 x + 36)}^{1} = {(x + 4)}^{2}$

Simplify.

$7 x + 36 = {(x + 4)}^{2}$

Simplify the right side.

Simplify ${(x + 4)}^{2}$ .

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$7 x + 36 = (x + 4) (x + 4)$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$7 x + 36 = x (x + 4) + 4 (x + 4)$

Apply the distributive property.

$7 x + 36 = x \cdot x + x \cdot 4 + 4 (x + 4)$

Apply the distributive property.

$7 x + 36 = x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$7 x + 36 = x^{2} + x \cdot 4 + 4 x + 4 \cdot 4$

Move $4$ to the left of $x$ .

$7 x + 36 = x^{2} + 4 \cdot x + 4 x + 4 \cdot 4$

Multiply $4$ by $4$ .

$7 x + 36 = x^{2} + 4 x + 4 x + 16$

Add $4 x$ and $4 x$ .

$7 x + 36 = x^{2} + 8 x + 16$

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.

$x^{2} + 8 x + 16 = 7 x + 36$

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

Subtract $7 x$ from both sides of the equation.

$x^{2} + 8 x + 16 - 7 x = 36$

Subtract $7 x$ from $8 x$ .

$x^{2} + x + 16 = 36$

Subtract $36$ from both sides of the equation.

$x^{2} + x + 16 - 36 = 0$

Subtract $36$ from $16$ .

$x^{2} + x - 20 = 0$

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$(x - 4) (x + 5) = 0$

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

$x - 4 = 0$

$x + 5 = 0$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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.

$x = - 5$

The final solution is all the values that make $(x - 4) (x + 5) = 0$ true.

$x = 4, - 5$

Step 6

Exclude the solutions that do not make $\sqrt{7 x + 36} - x - 4 = 0$ true.

$x = 4$

Solve by Factoring square root of 7x+36=x+4

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