Solve for x 2x = square root of 100-12x-2

$2 x = \sqrt{100 - 12 x} - 2$

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{100 - 12 x} - 2 = 2 x$

Add $2$ to both sides of the equation.

$\sqrt{100 - 12 x} = 2 x + 2$

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

${\sqrt{100 - 12 x}}^{2} = {(2 x + 2)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 - 12 x}$ as ${(100 - 12 x)}^{\frac{1}{2}}$ .

${({(100 - 12 x)}^{\frac{1}{2}})}^{2} = {(2 x + 2)}^{2}$

Simplify the left side.

Simplify ${({(100 - 12 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(100 - 12 x)}^{\frac{1}{2}})}^{2}$ .

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

${(100 - 12 x)}^{\frac{1}{2} \cdot 2} = {(2 x + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(100 - 12 x)}^{\frac{1}{2} \cdot 2} = {(2 x + 2)}^{2}$

Rewrite the expression.

${(100 - 12 x)}^{1} = {(2 x + 2)}^{2}$

Simplify.

$100 - 12 x = {(2 x + 2)}^{2}$

Simplify the right side.

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

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

$100 - 12 x = (2 x + 2) (2 x + 2)$

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

Apply the distributive property.

$100 - 12 x = 2 x (2 x + 2) + 2 (2 x + 2)$

Apply the distributive property.

$100 - 12 x = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x + 2)$

Apply the distributive property.

$100 - 12 x = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$100 - 12 x = 2 \cdot 2 x \cdot x + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

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

Move $x$ .

$100 - 12 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $x$ by $x$ .

$100 - 12 x = 2 \cdot 2 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$100 - 12 x = 4 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$100 - 12 x = 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$100 - 12 x = 4 x^{2} + 4 x + 4 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$100 - 12 x = 4 x^{2} + 4 x + 4 x + 4$

Add $4 x$ and $4 x$ .

$100 - 12 x = 4 x^{2} + 8 x + 4$

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.

$4 x^{2} + 8 x + 4 = 100 - 12 x$

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

Add $12 x$ to both sides of the equation.

$4 x^{2} + 8 x + 4 + 12 x = 100$

Add $8 x$ and $12 x$ .

$4 x^{2} + 20 x + 4 = 100$

Subtract $100$ from both sides of the equation.

$4 x^{2} + 20 x + 4 - 100 = 0$

Subtract $100$ from $4$ .

$4 x^{2} + 20 x - 96 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} + 20 x - 96$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) + 20 x - 96 = 0$

Factor $4$ out of $20 x$ .

$4 (x^{2}) + 4 (5 x) - 96 = 0$

Factor $4$ out of $- 96$ .

$4 x^{2} + 4 (5 x) + 4 \cdot - 24 = 0$

Factor $4$ out of $4 x^{2} + 4 (5 x)$ .

$4 (x^{2} + 5 x) + 4 \cdot - 24 = 0$

Factor $4$ out of $4 (x^{2} + 5 x) + 4 \cdot - 24$ .

$4 (x^{2} + 5 x - 24) = 0$

Factor.

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$4 ((x - 3) (x + 8)) = 0$

Remove unnecessary parentheses.

$4 (x - 3) (x + 8) = 0$

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

$x - 3 = 0$

$x + 8 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $4 (x - 3) (x + 8) = 0$ true.

$x = 3, - 8$

Exclude the solutions that do not make $2 x = \sqrt{100 - 12 x} - 2$ true.

$x = 3$

Solve for x 2x = square root of 100-12x-2

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