Solve for w 6+3w = square root of 2w+12+2w

$6 + 3 w = \sqrt{2 w + 12} + 2 w$

Step 1

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

$\sqrt{2 w + 12} + 2 w = 6 + 3 w$

Step 2

Move all terms not containing $\sqrt{2 w + 12}$ to the right side of the equation.

Subtract $2 w$ from both sides of the equation.

$\sqrt{2 w + 12} = 6 + 3 w - 2 w$

Subtract $2 w$ from $3 w$ .

$\sqrt{2 w + 12} = 6 + w$

Step 3

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

${\sqrt{2 w + 12}}^{2} = {(6 + w)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(2 w + 12)}^{\frac{1}{2}})}^{2} = {(6 + w)}^{2}$

Simplify the left side.

Simplify ${({(2 w + 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 w + 12)}^{\frac{1}{2}})}^{2}$ .

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

${(2 w + 12)}^{\frac{1}{2} \cdot 2} = {(6 + w)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 w + 12)}^{\frac{1}{2} \cdot 2} = {(6 + w)}^{2}$

Rewrite the expression.

${(2 w + 12)}^{1} = {(6 + w)}^{2}$

Simplify.

$2 w + 12 = {(6 + w)}^{2}$

Simplify the right side.

Simplify ${(6 + w)}^{2}$ .

Rewrite ${(6 + w)}^{2}$ as $(6 + w) (6 + w)$ .

$2 w + 12 = (6 + w) (6 + w)$

Expand $(6 + w) (6 + w)$ using the FOIL Method.

Apply the distributive property.

$2 w + 12 = 6 (6 + w) + w (6 + w)$

Apply the distributive property.

$2 w + 12 = 6 \cdot 6 + 6 w + w (6 + w)$

Apply the distributive property.

$2 w + 12 = 6 \cdot 6 + 6 w + w \cdot 6 + w \cdot w$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$2 w + 12 = 36 + 6 w + w \cdot 6 + w \cdot w$

Move $6$ to the left of $w$ .

$2 w + 12 = 36 + 6 w + 6 \cdot w + w \cdot w$

Multiply $w$ by $w$ .

$2 w + 12 = 36 + 6 w + 6 w + w^{2}$

Add $6 w$ and $6 w$ .

$2 w + 12 = 36 + 12 w + w^{2}$

Step 5

Solve for $w$ .

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

$36 + 12 w + w^{2} = 2 w + 12$

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

Subtract $2 w$ from both sides of the equation.

$36 + 12 w + w^{2} - 2 w = 12$

Subtract $2 w$ from $12 w$ .

$36 + w^{2} + 10 w = 12$

Subtract $12$ from both sides of the equation.

$36 + w^{2} + 10 w - 12 = 0$

Subtract $12$ from $36$ .

$w^{2} + 10 w + 24 = 0$

Factor $w^{2} + 10 w + 24$ 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 $10$ .

$4, 6$

Write the factored form using these integers.

$(w + 4) (w + 6) = 0$

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

$w + 4 = 0$

$w + 6 = 0$

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

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

$w + 4 = 0$

Subtract $4$ from both sides of the equation.

$w = - 4$

Set $w + 6$ equal to $0$ and solve for $w$ .

Set $w + 6$ equal to $0$ .

$w + 6 = 0$

Subtract $6$ from both sides of the equation.

$w = - 6$

The final solution is all the values that make $(w + 4) (w + 6) = 0$ true.

$w = - 4, - 6$

Solve for w 6+3w = square root of 2w+12+2w

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