Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{3}{4}$ and $x^{2}$ .

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

Step 1.1.2

Combine $x$ and $\frac{1}{4}$ .

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

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.2.3.2

Factor $2$ out of $4$ .

$3 x^{2} - x + 2 (2) (\frac{- 1}{2}) = 0$

Step 2.2.3.3

Cancel the common factor.

$3 x^{2} - x + 2 \cdot (2 (\frac{- 1}{2})) = 0$

Step 2.2.3.4

Rewrite the expression.

$3 x^{2} - x + 2 \cdot - 1 = 0$

Step 2.2.4

Multiply $2$ by $- 1$ .

$3 x^{2} - x - 2 = 0$

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 $a = 3$ , $b = - 1$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (3 \cdot - 2)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 2}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{1 \pm \sqrt{1 - 12 \cdot - 2}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{1 \pm \sqrt{1 + 24}}{2 \cdot 3}$

Step 5.1.3

Add $1$ and $24$ .

$x = \frac{1 \pm \sqrt{25}}{2 \cdot 3}$

Step 5.1.4

Rewrite $25$ as $5^{2}$ .

$x = \frac{1 \pm \sqrt{5^{2}}}{2 \cdot 3}$

Step 5.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{1 \pm 5}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{1 \pm 5}{6}$

Step 6

The final answer is the combination of both solutions.

$x = 1, - \frac{2}{3}$