Solve for w 5w- square root of 27w+7=3w-1

$5 w - \sqrt{27 w + 7} = 3 w - 1$

Step 1

Move all terms not containing $- \sqrt{27 w + 7}$ to the right side of the equation.

Subtract $5 w$ from both sides of the equation.

$- \sqrt{27 w + 7} = 3 w - 1 - 5 w$

Subtract $5 w$ from $3 w$ .

$- \sqrt{27 w + 7} = - 2 w - 1$

Step 2

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

${(- \sqrt{27 w + 7})}^{2} = {(- 2 w - 1)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 w + 7}$ as ${(27 w + 7)}^{\frac{1}{2}}$ .

${(- {(27 w + 7)}^{\frac{1}{2}})}^{2} = {(- 2 w - 1)}^{2}$

Simplify the left side.

Simplify ${(- {(27 w + 7)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(27 w + 7)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(27 w + 7)}^{\frac{1}{2}})}^{2} = {(- 2 w - 1)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(27 w + 7)}^{\frac{1}{2}})}^{2} = {(- 2 w - 1)}^{2}$

Multiply ${({(27 w + 7)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(27 w + 7)}^{\frac{1}{2}})}^{2} = {(- 2 w - 1)}^{2}$

Multiply the exponents in ${({(27 w + 7)}^{\frac{1}{2}})}^{2}$ .

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

${(27 w + 7)}^{\frac{1}{2} \cdot 2} = {(- 2 w - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(27 w + 7)}^{\frac{1}{2} \cdot 2} = {(- 2 w - 1)}^{2}$

Rewrite the expression.

${(27 w + 7)}^{1} = {(- 2 w - 1)}^{2}$

Simplify.

$27 w + 7 = {(- 2 w - 1)}^{2}$

Simplify the right side.

Simplify ${(- 2 w - 1)}^{2}$ .

Rewrite ${(- 2 w - 1)}^{2}$ as $(- 2 w - 1) (- 2 w - 1)$ .

$27 w + 7 = (- 2 w - 1) (- 2 w - 1)$

Expand $(- 2 w - 1) (- 2 w - 1)$ using the FOIL Method.

Apply the distributive property.

$27 w + 7 = - 2 w (- 2 w - 1) - 1 (- 2 w - 1)$

Apply the distributive property.

$27 w + 7 = - 2 w (- 2 w) - 2 w \cdot - 1 - 1 (- 2 w - 1)$

Apply the distributive property.

$27 w + 7 = - 2 w (- 2 w) - 2 w \cdot - 1 - 1 (- 2 w) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$27 w + 7 = - 2 \cdot - 2 w \cdot w - 2 w \cdot - 1 - 1 (- 2 w) - 1 \cdot - 1$

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

Move $w$ .

$27 w + 7 = - 2 \cdot - 2 (w \cdot w) - 2 w \cdot - 1 - 1 (- 2 w) - 1 \cdot - 1$

Multiply $w$ by $w$ .

$27 w + 7 = - 2 \cdot - 2 w^{2} - 2 w \cdot - 1 - 1 (- 2 w) - 1 \cdot - 1$

Multiply $- 2$ by $- 2$ .

$27 w + 7 = 4 w^{2} - 2 w \cdot - 1 - 1 (- 2 w) - 1 \cdot - 1$

Multiply $- 1$ by $- 2$ .

$27 w + 7 = 4 w^{2} + 2 w - 1 (- 2 w) - 1 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$27 w + 7 = 4 w^{2} + 2 w + 2 w - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$27 w + 7 = 4 w^{2} + 2 w + 2 w + 1$

Add $2 w$ and $2 w$ .

$27 w + 7 = 4 w^{2} + 4 w + 1$

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.

$4 w^{2} + 4 w + 1 = 27 w + 7$

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

Subtract $27 w$ from both sides of the equation.

$4 w^{2} + 4 w + 1 - 27 w = 7$

Subtract $27 w$ from $4 w$ .

$4 w^{2} - 23 w + 1 = 7$

Subtract $7$ from both sides of the equation.

$4 w^{2} - 23 w + 1 - 7 = 0$

Subtract $7$ from $1$ .

$4 w^{2} - 23 w - 6 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 w$ .

$4 w^{2} - 23 w - 6 = 0$

Rewrite $- 23$ as $1$ plus $- 24$

$4 w^{2} + (1 - 24) w - 6 = 0$

Apply the distributive property.

$4 w^{2} + 1 w - 24 w - 6 = 0$

Multiply $w$ by $1$ .

$4 w^{2} + w - 24 w - 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 w^{2} + w) - 24 w - 6 = 0$

Factor out the greatest common factor (GCF) from each group.

$w (4 w + 1) - 6 (4 w + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 w + 1$ .

$(4 w + 1) (w - 6) = 0$

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

$4 w + 1 = 0$

$w - 6 = 0$

Set $4 w + 1$ equal to $0$ and solve for $w$ .

Set $4 w + 1$ equal to $0$ .

$4 w + 1 = 0$

Solve $4 w + 1 = 0$ for $w$ .

Subtract $1$ from both sides of the equation.

$4 w = - 1$

Divide each term in $4 w = - 1$ by $4$ and simplify.

Divide each term in $4 w = - 1$ by $4$ .

$\frac{4 w}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 w}{4} = \frac{- 1}{4}$

Divide $w$ by $1$ .

$w = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$w = - \frac{1}{4}$

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

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

$w - 6 = 0$

Add $6$ to both sides of the equation.

$w = 6$

The final solution is all the values that make $(4 w + 1) (w - 6) = 0$ true.

$w = - \frac{1}{4}, 6$

Step 5

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{1}{4}, 6$

Decimal Form:

$w = - 0.25, 6$

Solve for w 5w- square root of 27w+7=3w-1

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