Solve for w square root of -22w-6+w=3

$\sqrt{- 22 w - 6} + w = 3$

Step 1

Subtract $w$ from both sides of the equation.

$\sqrt{- 22 w - 6} = 3 - w$

Step 2

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

${\sqrt{- 22 w - 6}}^{2} = {(3 - w)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 22 w - 6}$ as ${(- 22 w - 6)}^{\frac{1}{2}}$ .

${({(- 22 w - 6)}^{\frac{1}{2}})}^{2} = {(3 - w)}^{2}$

Simplify the left side.

Simplify ${({(- 22 w - 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- 22 w - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(- 22 w - 6)}^{\frac{1}{2} \cdot 2} = {(3 - w)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 22 w - 6)}^{\frac{1}{2} \cdot 2} = {(3 - w)}^{2}$

Rewrite the expression.

${(- 22 w - 6)}^{1} = {(3 - w)}^{2}$

Simplify.

$- 22 w - 6 = {(3 - w)}^{2}$

Simplify the right side.

Simplify ${(3 - w)}^{2}$ .

Rewrite ${(3 - w)}^{2}$ as $(3 - w) (3 - w)$ .

$- 22 w - 6 = (3 - w) (3 - w)$

Expand $(3 - w) (3 - w)$ using the FOIL Method.

Apply the distributive property.

$- 22 w - 6 = 3 (3 - w) - w (3 - w)$

Apply the distributive property.

$- 22 w - 6 = 3 \cdot 3 + 3 (- w) - w (3 - w)$

Apply the distributive property.

$- 22 w - 6 = 3 \cdot 3 + 3 (- w) - w \cdot 3 - w (- w)$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$- 22 w - 6 = 9 + 3 (- w) - w \cdot 3 - w (- w)$

Multiply $- 1$ by $3$ .

$- 22 w - 6 = 9 - 3 w - w \cdot 3 - w (- w)$

Multiply $3$ by $- 1$ .

$- 22 w - 6 = 9 - 3 w - 3 w - w (- w)$

Rewrite using the commutative property of multiplication.

$- 22 w - 6 = 9 - 3 w - 3 w - 1 \cdot - 1 w \cdot w$

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$- 22 w - 6 = 9 - 3 w - 3 w - 1 \cdot - 1 (w \cdot w)$

Multiply $w$ by $w$ .

$- 22 w - 6 = 9 - 3 w - 3 w - 1 \cdot - 1 w^{2}$

Multiply $- 1$ by $- 1$ .

$- 22 w - 6 = 9 - 3 w - 3 w + 1 w^{2}$

Multiply $w^{2}$ by $1$ .

$- 22 w - 6 = 9 - 3 w - 3 w + w^{2}$

Subtract $3 w$ from $- 3 w$ .

$- 22 w - 6 = 9 - 6 w + w^{2}$

Step 4

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.

$9 - 6 w + w^{2} = - 22 w - 6$

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

Add $22 w$ to both sides of the equation.

$9 - 6 w + w^{2} + 22 w = - 6$

Add $- 6 w$ and $22 w$ .

$9 + w^{2} + 16 w = - 6$

Add $6$ to both sides of the equation.

$9 + w^{2} + 16 w + 6 = 0$

Add $9$ and $6$ .

$w^{2} + 16 w + 15 = 0$

Factor $w^{2} + 16 w + 15$ 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 $15$ and whose sum is $16$ .

$1, 15$

Write the factored form using these integers.

$(w + 1) (w + 15) = 0$

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

$w + 1 = 0$

$w + 15 = 0$

Set $w + 1$ equal to $0$ and solve for $w$ .

Set $w + 1$ equal to $0$ .

$w + 1 = 0$

Subtract $1$ from both sides of the equation.

$w = - 1$

Set $w + 15$ equal to $0$ and solve for $w$ .

Set $w + 15$ equal to $0$ .

$w + 15 = 0$

Subtract $15$ from both sides of the equation.

$w = - 15$

The final solution is all the values that make $(w + 1) (w + 15) = 0$ true.

$w = - 1, - 15$

Solve for w square root of -22w-6+w=3

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