Solve by Factoring square root of 54-x=x+2

$\sqrt{54 - x} = x + 2$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{54 - x} - x = 2$

Subtract $2$ from both sides of the equation.

$\sqrt{54 - x} - x - 2 = 0$

Step 2

Move all terms not containing $\sqrt{54 - x}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{54 - x} - 2 = x$

Add $2$ to both sides of the equation.

$\sqrt{54 - x} = x + 2$

Step 3

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

${\sqrt{54 - x}}^{2} = {(x + 2)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{54 - x}$ as ${(54 - x)}^{\frac{1}{2}}$ .

${({(54 - x)}^{\frac{1}{2}})}^{2} = {(x + 2)}^{2}$

Simplify the left side.

Simplify ${({(54 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(54 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(54 - x)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(54 - x)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Rewrite the expression.

${(54 - x)}^{1} = {(x + 2)}^{2}$

Simplify.

$54 - x = {(x + 2)}^{2}$

Simplify the right side.

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

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

$54 - x = (x + 2) (x + 2)$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$54 - x = x (x + 2) + 2 (x + 2)$

Apply the distributive property.

$54 - x = x \cdot x + x \cdot 2 + 2 (x + 2)$

Apply the distributive property.

$54 - x = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$54 - x = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Move $2$ to the left of $x$ .

$54 - x = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$54 - x = x^{2} + 2 x + 2 x + 4$

Add $2 x$ and $2 x$ .

$54 - x = x^{2} + 4 x + 4$

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.

$x^{2} + 4 x + 4 = 54 - x$

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

Add $x$ to both sides of the equation.

$x^{2} + 4 x + 4 + x = 54$

Add $4 x$ and $x$ .

$x^{2} + 5 x + 4 = 54$

Subtract $54$ from both sides of the equation.

$x^{2} + 5 x + 4 - 54 = 0$

Subtract $54$ from $4$ .

$x^{2} + 5 x - 50 = 0$

Factor $x^{2} + 5 x - 50$ 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 $- 50$ and whose sum is $5$ .

$- 5, 10$

Write the factored form using these integers.

$(x - 5) (x + 10) = 0$

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

$x - 5 = 0$

$x + 10 = 0$

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

The final solution is all the values that make $(x - 5) (x + 10) = 0$ true.

$x = 5, - 10$

Step 6

Exclude the solutions that do not make $\sqrt{54 - x} - x - 2 = 0$ true.

$x = 5$

Solve by Factoring square root of 54-x=x+2

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