Solve by Factoring square root of 3x+24-2=x

$\sqrt{3 x + 24} - 2 = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{3 x + 24} - 2 - x = 0$

Step 2

Factor $3$ out of $3 x + 24$ .

Factor $3$ out of $3 x$ .

$\sqrt{3 (x) + 24} - 2 - x = 0$

Factor $3$ out of $24$ .

$\sqrt{3 x + 3 \cdot 8} - 2 - x = 0$

Factor $3$ out of $3 x + 3 \cdot 8$ .

$\sqrt{3 (x + 8)} - 2 - x = 0$

Step 3

Move all terms not containing $\sqrt{3 (x + 8)}$ to the right side of the equation.

Add $2$ to both sides of the equation.

$\sqrt{3 (x + 8)} - x = 2$

Add $x$ to both sides of the equation.

$\sqrt{3 (x + 8)} = 2 + x$

Step 4

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

${\sqrt{3 (x + 8)}}^{2} = {(2 + x)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (x + 8)}$ as ${(3 (x + 8))}^{\frac{1}{2}}$ .

${({(3 (x + 8))}^{\frac{1}{2}})}^{2} = {(2 + x)}^{2}$

Simplify the left side.

Simplify ${({(3 (x + 8))}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 (x + 8))}^{\frac{1}{2}})}^{2}$ .

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

${(3 (x + 8))}^{\frac{1}{2} \cdot 2} = {(2 + x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 (x + 8))}^{\frac{1}{2} \cdot 2} = {(2 + x)}^{2}$

Rewrite the expression.

${(3 (x + 8))}^{1} = {(2 + x)}^{2}$

Apply the distributive property.

${(3 x + 3 \cdot 8)}^{1} = {(2 + x)}^{2}$

Multiply.

Multiply $3$ by $8$ .

${(3 x + 24)}^{1} = {(2 + x)}^{2}$

Simplify.

$3 x + 24 = {(2 + x)}^{2}$

Simplify the right side.

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

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

$3 x + 24 = (2 + x) (2 + x)$

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

Apply the distributive property.

$3 x + 24 = 2 (2 + x) + x (2 + x)$

Apply the distributive property.

$3 x + 24 = 2 \cdot 2 + 2 x + x (2 + x)$

Apply the distributive property.

$3 x + 24 = 2 \cdot 2 + 2 x + x \cdot 2 + x \cdot x$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$3 x + 24 = 4 + 2 x + x \cdot 2 + x \cdot x$

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

$3 x + 24 = 4 + 2 x + 2 \cdot x + x \cdot x$

Multiply $x$ by $x$ .

$3 x + 24 = 4 + 2 x + 2 x + x^{2}$

Add $2 x$ and $2 x$ .

$3 x + 24 = 4 + 4 x + x^{2}$

Step 6

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.

$4 + 4 x + x^{2} = 3 x + 24$

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

Subtract $3 x$ from both sides of the equation.

$4 + 4 x + x^{2} - 3 x = 24$

Subtract $3 x$ from $4 x$ .

$4 + x^{2} + x = 24$

Subtract $24$ from both sides of the equation.

$4 + x^{2} + x - 24 = 0$

Subtract $24$ from $4$ .

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

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 7

Exclude the solutions that do not make $\sqrt{3 (x + 8)} - 2 - x = 0$ true.

$x = 4$

Solve by Factoring square root of 3x+24-2=x

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