Algebra Examples

$\frac{1}{2} w^{2} - \frac{1}{4} = \frac{3}{16} w$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Combine $\frac{1}{2}$ and $w^{2}$ .

$\frac{w^{2}}{2} - \frac{1}{4} = \frac{3}{16} w$

Step 1.2

Simplify the right side.

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Step 1.2.1

Combine $\frac{3}{16}$ and $w$ .

$\frac{w^{2}}{2} - \frac{1}{4} = \frac{3 w}{16}$

Step 1.3

Subtract $\frac{3 w}{16}$ from both sides of the equation.

$\frac{w^{2}}{2} - \frac{1}{4} - \frac{3 w}{16} = 0$

Step 2

Multiply through by the least common denominator $16$ , then simplify.

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Step 2.1

Apply the distributive property.

$16 (\frac{w^{2}}{2}) + 16 (- \frac{1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Factor $2$ out of $16$ .

$2 (8) (\frac{w^{2}}{2}) + 16 (- \frac{1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.1.2

Cancel the common factor.

$2 \cdot (8 (\frac{w^{2}}{2})) + 16 (- \frac{1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.1.3

Rewrite the expression.

$8 w^{2} + 16 (- \frac{1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$8 w^{2} + 16 (\frac{- 1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.2.2

Factor $4$ out of $16$ .

$8 w^{2} + 4 (4) (\frac{- 1}{4}) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.2.3

Cancel the common factor.

$8 w^{2} + 4 \cdot (4 (\frac{- 1}{4})) + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.2.4

Rewrite the expression.

$8 w^{2} + 4 \cdot - 1 + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.3

Multiply $4$ by $- 1$ .

$8 w^{2} - 4 + 16 (- \frac{3 w}{16}) = 0$

Step 2.2.4

Cancel the common factor of $16$ .

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Step 2.2.4.1

Move the leading negative in $- \frac{3 w}{16}$ into the numerator.

$8 w^{2} - 4 + 16 (\frac{- 3 w}{16}) = 0$

Step 2.2.4.2

Cancel the common factor.

$8 w^{2} - 4 + 16 (\frac{- 3 w}{16}) = 0$

Step 2.2.4.3

Rewrite the expression.

$8 w^{2} - 4 - 3 w = 0$

Step 2.3

Move $- 4$ .

$8 w^{2} - 3 w - 4 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 8$ , $b = - 3$ , and $c = - 4$ into the quadratic formula and solve for $w$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (8 \cdot - 4)}}{2 \cdot 8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 3$ to the power of $2$ .

$w = \frac{3 \pm \sqrt{9 - 4 \cdot 8 \cdot - 4}}{2 \cdot 8}$

Step 5.1.2

Multiply $- 4 \cdot 8 \cdot - 4$ .

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Step 5.1.2.1

Multiply $- 4$ by $8$ .

$w = \frac{3 \pm \sqrt{9 - 32 \cdot - 4}}{2 \cdot 8}$

Step 5.1.2.2

Multiply $- 32$ by $- 4$ .

$w = \frac{3 \pm \sqrt{9 + 128}}{2 \cdot 8}$

Step 5.1.3

Add $9$ and $128$ .

$w = \frac{3 \pm \sqrt{137}}{2 \cdot 8}$

Step 5.2

Multiply $2$ by $8$ .

$w = \frac{3 \pm \sqrt{137}}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = \frac{3 \pm \sqrt{137}}{16}$

Decimal Form:

$w = 0.91904374 \dots, - 0.54404374 \dots$