Solve by Factoring square root of 3x+28-8=x

$\sqrt{3 x + 28} - 8 = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{3 x + 28} - 8 - x = 0$

Step 2

Move all terms not containing $\sqrt{3 x + 28}$ to the right side of the equation.

Add $8$ to both sides of the equation.

$\sqrt{3 x + 28} - x = 8$

Add $x$ to both sides of the equation.

$\sqrt{3 x + 28} = 8 + x$

Step 3

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

${\sqrt{3 x + 28}}^{2} = {(8 + x)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 28}$ as ${(3 x + 28)}^{\frac{1}{2}}$ .

${({(3 x + 28)}^{\frac{1}{2}})}^{2} = {(8 + x)}^{2}$

Simplify the left side.

Simplify ${({(3 x + 28)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 28)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x + 28)}^{\frac{1}{2} \cdot 2} = {(8 + x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x + 28)}^{\frac{1}{2} \cdot 2} = {(8 + x)}^{2}$

Rewrite the expression.

${(3 x + 28)}^{1} = {(8 + x)}^{2}$

Simplify.

$3 x + 28 = {(8 + x)}^{2}$

Simplify the right side.

Simplify ${(8 + x)}^{2}$ .

Rewrite ${(8 + x)}^{2}$ as $(8 + x) (8 + x)$ .

$3 x + 28 = (8 + x) (8 + x)$

Expand $(8 + x) (8 + x)$ using the FOIL Method.

Apply the distributive property.

$3 x + 28 = 8 (8 + x) + x (8 + x)$

Apply the distributive property.

$3 x + 28 = 8 \cdot 8 + 8 x + x (8 + x)$

Apply the distributive property.

$3 x + 28 = 8 \cdot 8 + 8 x + x \cdot 8 + x \cdot x$

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

$3 x + 28 = 64 + 8 x + x \cdot 8 + x \cdot x$

Move $8$ to the left of $x$ .

$3 x + 28 = 64 + 8 x + 8 \cdot x + x \cdot x$

Multiply $x$ by $x$ .

$3 x + 28 = 64 + 8 x + 8 x + x^{2}$

Add $8 x$ and $8 x$ .

$3 x + 28 = 64 + 16 x + x^{2}$

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.

$64 + 16 x + x^{2} = 3 x + 28$

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

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

$64 + 16 x + x^{2} - 3 x = 28$

Subtract $3 x$ from $16 x$ .

$64 + x^{2} + 13 x = 28$

Subtract $28$ from both sides of the equation.

$64 + x^{2} + 13 x - 28 = 0$

Subtract $28$ from $64$ .

$x^{2} + 13 x + 36 = 0$

Factor $x^{2} + 13 x + 36$ 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 $36$ and whose sum is $13$ .

$4, 9$

Write the factored form using these integers.

$(x + 4) (x + 9) = 0$

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

$x + 4 = 0$

$x + 9 = 0$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $(x + 4) (x + 9) = 0$ true.

$x = - 4, - 9$

Step 6

Exclude the solutions that do not make $\sqrt{3 x + 28} - 8 - x = 0$ true.

$x = - 4$

Solve by Factoring square root of 3x+28-8=x

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