Precalculus Examples

$w = \frac{\sqrt{1 - 3 w}}{2}$

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.

$\frac{\sqrt{1 - 3 w}}{2} = w$

Step 2

Multiply both sides by $2$ .

$\frac{\sqrt{1 - 3 w}}{2} \cdot 2 = w \cdot 2$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{\sqrt{1 - 3 w}}{2} \cdot 2 = w \cdot 2$

Step 3.1.1.2

Rewrite the expression.

$\sqrt{1 - 3 w} = w \cdot 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Move $2$ to the left of $w$ .

$\sqrt{1 - 3 w} = 2 w$

Step 4

Solve for $w$ .

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Step 4.1

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

${\sqrt{1 - 3 w}}^{2} = {(2 w)}^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 3 w}$ as ${(1 - 3 w)}^{\frac{1}{2}}$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(1 - 3 w)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(1 - 3 w)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

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

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.2.1.2

Simplify.

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

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Step 4.2.3.1.1

Apply the product rule to $2 w$ .

$1 - 3 w = 2^{2} w^{2}$

Step 4.2.3.1.2

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

$1 - 3 w = 4 w^{2}$

Step 4.3

Solve for $w$ .

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Step 4.3.1

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

$1 - 3 w - 4 w^{2} = 0$

Step 4.3.2

Factor the left side of the equation.

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Step 4.3.2.1

Factor $- 1$ out of $1 - 3 w - 4 w^{2}$ .

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Step 4.3.2.1.1

Reorder the expression.

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Step 4.3.2.1.1.1

Move $1$ .

$- 3 w - 4 w^{2} + 1 = 0$

Step 4.3.2.1.1.2

Reorder $- 3 w$ and $- 4 w^{2}$ .

$- 4 w^{2} - 3 w + 1 = 0$

Step 4.3.2.1.2

Factor $- 1$ out of $- 4 w^{2}$ .

$- (4 w^{2}) - 3 w + 1 = 0$

Step 4.3.2.1.3

Factor $- 1$ out of $- 3 w$ .

$- (4 w^{2}) - (3 w) + 1 = 0$

Step 4.3.2.1.4

Rewrite $1$ as $- 1 (- 1)$ .

$- (4 w^{2}) - (3 w) - 1 \cdot - 1 = 0$

Step 4.3.2.1.5

Factor $- 1$ out of $- (4 w^{2}) - (3 w)$ .

$- (4 w^{2} + 3 w) - 1 \cdot - 1 = 0$

Step 4.3.2.1.6

Factor $- 1$ out of $- (4 w^{2} + 3 w) - 1 (- 1)$ .

$- (4 w^{2} + 3 w - 1) = 0$

Step 4.3.2.2

Factor.

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Step 4.3.2.2.1

Factor by grouping.

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Step 4.3.2.2.1.1

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 $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = 3$ .

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Step 4.3.2.2.1.1.1

Factor $3$ out of $3 w$ .

$- (4 w^{2} + 3 (w) - 1) = 0$

Step 4.3.2.2.1.1.2

Rewrite $3$ as $- 1$ plus $4$

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

Step 4.3.2.2.1.1.3

Apply the distributive property.

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

Step 4.3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.3.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.3.2.2.1.2.2

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

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

Step 4.3.2.2.1.3

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

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

Step 4.3.2.2.2

Remove unnecessary parentheses.

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

Step 4.3.3

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

$4 w - 1 = 0$

$w + 1 = 0$

Step 4.3.4

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

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Step 4.3.4.1

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

$4 w - 1 = 0$

Step 4.3.4.2

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

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Step 4.3.4.2.1

Add $1$ to both sides of the equation.

$4 w = 1$

Step 4.3.4.2.2

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

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Step 4.3.4.2.2.1

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

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

Step 4.3.4.2.2.2

Simplify the left side.

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Step 4.3.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.3.4.2.2.2.1.1

Cancel the common factor.

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

Step 4.3.4.2.2.2.1.2

Divide $w$ by $1$ .

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

Step 4.3.5

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

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Step 4.3.5.1

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

$w + 1 = 0$

Step 4.3.5.2

Subtract $1$ from both sides of the equation.

$w = - 1$

Step 4.3.6

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

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

Step 5

Exclude the solutions that do not make $w = \frac{\sqrt{1 - 3 w}}{2}$ true.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$w = 0.25$