Solve by Factoring square root of 6x-x+12=0

$\sqrt{6 x} - x + 12 = 0$

Step 1

Move all terms not containing $\sqrt{6 x}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{6 x} + 12 = x$

Subtract $12$ from both sides of the equation.

$\sqrt{6 x} = x - 12$

Step 2

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

${\sqrt{6 x}}^{2} = {(x - 12)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x}$ as ${(6 x)}^{\frac{1}{2}}$ .

${({(6 x)}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

Simplify the left side.

Simplify ${({(6 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 x)}^{\frac{1}{2}})}^{2}$ .

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

${(6 x)}^{\frac{1}{2} \cdot 2} = {(x - 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 x)}^{\frac{1}{2} \cdot 2} = {(x - 12)}^{2}$

Rewrite the expression.

${(6 x)}^{1} = {(x - 12)}^{2}$

Simplify.

$6 x = {(x - 12)}^{2}$

Simplify the right side.

Simplify ${(x - 12)}^{2}$ .

Rewrite ${(x - 12)}^{2}$ as $(x - 12) (x - 12)$ .

$6 x = (x - 12) (x - 12)$

Expand $(x - 12) (x - 12)$ using the FOIL Method.

Apply the distributive property.

$6 x = x (x - 12) - 12 (x - 12)$

Apply the distributive property.

$6 x = x \cdot x + x \cdot - 12 - 12 (x - 12)$

Apply the distributive property.

$6 x = x \cdot x + x \cdot - 12 - 12 x - 12 \cdot - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$6 x = x^{2} + x \cdot - 12 - 12 x - 12 \cdot - 12$

Move $- 12$ to the left of $x$ .

$6 x = x^{2} - 12 \cdot x - 12 x - 12 \cdot - 12$

Multiply $- 12$ by $- 12$ .

$6 x = x^{2} - 12 x - 12 x + 144$

Subtract $12 x$ from $- 12 x$ .

$6 x = x^{2} - 24 x + 144$

Step 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.

$x^{2} - 24 x + 144 = 6 x$

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

Subtract $6 x$ from both sides of the equation.

$x^{2} - 24 x + 144 - 6 x = 0$

Subtract $6 x$ from $- 24 x$ .

$x^{2} - 30 x + 144 = 0$

Factor $x^{2} - 30 x + 144$ 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 $144$ and whose sum is $- 30$ .

$- 24, - 6$

Write the factored form using these integers.

$(x - 24) (x - 6) = 0$

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

$x - 24 = 0$

$x - 6 = 0$

Set $x - 24$ equal to $0$ and solve for $x$ .

Set $x - 24$ equal to $0$ .

$x - 24 = 0$

Add $24$ to both sides of the equation.

$x = 24$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(x - 24) (x - 6) = 0$ true.

$x = 24, 6$

Step 5

Exclude the solutions that do not make $\sqrt{6 x} - x + 12 = 0$ true.

$x = 24$

Solve by Factoring square root of 6x-x+12=0

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