Solve for w 2w+3 = square root of 4w^2+14w+12

$2 w + 3 = \sqrt{4 w^{2} + 14 w + 12}$

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.

$\sqrt{4 w^{2} + 14 w + 12} = 2 w + 3$

Step 2

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

${\sqrt{4 w^{2} + 14 w + 12}}^{2} = {(2 w + 3)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 w^{2} + 14 w + 12}$ as ${(4 w^{2} + 14 w + 12)}^{\frac{1}{2}}$ .

${({(4 w^{2} + 14 w + 12)}^{\frac{1}{2}})}^{2} = {(2 w + 3)}^{2}$

Simplify the left side.

Simplify ${({(4 w^{2} + 14 w + 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 w^{2} + 14 w + 12)}^{\frac{1}{2}})}^{2}$ .

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

${(4 w^{2} + 14 w + 12)}^{\frac{1}{2} \cdot 2} = {(2 w + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 w^{2} + 14 w + 12)}^{\frac{1}{2} \cdot 2} = {(2 w + 3)}^{2}$

Rewrite the expression.

${(4 w^{2} + 14 w + 12)}^{1} = {(2 w + 3)}^{2}$

Simplify.

$4 w^{2} + 14 w + 12 = {(2 w + 3)}^{2}$

Simplify the right side.

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

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

$4 w^{2} + 14 w + 12 = (2 w + 3) (2 w + 3)$

Expand $(2 w + 3) (2 w + 3)$ using the FOIL Method.

Apply the distributive property.

$4 w^{2} + 14 w + 12 = 2 w (2 w + 3) + 3 (2 w + 3)$

Apply the distributive property.

$4 w^{2} + 14 w + 12 = 2 w (2 w) + 2 w \cdot 3 + 3 (2 w + 3)$

Apply the distributive property.

$4 w^{2} + 14 w + 12 = 2 w (2 w) + 2 w \cdot 3 + 3 (2 w) + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 w^{2} + 14 w + 12 = 2 \cdot 2 w \cdot w + 2 w \cdot 3 + 3 (2 w) + 3 \cdot 3$

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

Move $w$ .

$4 w^{2} + 14 w + 12 = 2 \cdot 2 (w \cdot w) + 2 w \cdot 3 + 3 (2 w) + 3 \cdot 3$

Multiply $w$ by $w$ .

$4 w^{2} + 14 w + 12 = 2 \cdot 2 w^{2} + 2 w \cdot 3 + 3 (2 w) + 3 \cdot 3$

Multiply $2$ by $2$ .

$4 w^{2} + 14 w + 12 = 4 w^{2} + 2 w \cdot 3 + 3 (2 w) + 3 \cdot 3$

Multiply $3$ by $2$ .

$4 w^{2} + 14 w + 12 = 4 w^{2} + 6 w + 3 (2 w) + 3 \cdot 3$

Multiply $2$ by $3$ .

$4 w^{2} + 14 w + 12 = 4 w^{2} + 6 w + 6 w + 3 \cdot 3$

Multiply $3$ by $3$ .

$4 w^{2} + 14 w + 12 = 4 w^{2} + 6 w + 6 w + 9$

Add $6 w$ and $6 w$ .

$4 w^{2} + 14 w + 12 = 4 w^{2} + 12 w + 9$

Step 4

Solve for $w$ .

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

Subtract $4 w^{2}$ from both sides of the equation.

$4 w^{2} + 14 w + 12 - 4 w^{2} = 12 w + 9$

Subtract $12 w$ from both sides of the equation.

$4 w^{2} + 14 w + 12 - 4 w^{2} - 12 w = 9$

Combine the opposite terms in $4 w^{2} + 14 w + 12 - 4 w^{2} - 12 w$ .

Subtract $4 w^{2}$ from $4 w^{2}$ .

$14 w + 12 + 0 - 12 w = 9$

Add $14 w + 12$ and $0$ .

$14 w + 12 - 12 w = 9$

Subtract $12 w$ from $14 w$ .

$2 w + 12 = 9$

Move all terms not containing $w$ to the right side of the equation.

Subtract $12$ from both sides of the equation.

$2 w = 9 - 12$

Subtract $12$ from $9$ .

$2 w = - 3$

Divide each term in $2 w = - 3$ by $2$ and simplify.

Divide each term in $2 w = - 3$ by $2$ .

$\frac{2 w}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 w}{2} = \frac{- 3}{2}$

Divide $w$ by $1$ .

$w = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$w = - \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{3}{2}$

Decimal Form:

$w = - 1.5$

Mixed Number Form:

$w = - 1 \frac{1}{2}$

Solve for w 2w+3 = square root of 4w^2+14w+12

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