Algebra Examples

$3 x = \frac{1}{4} x^{2} + 5$

Step 1

Combine $\frac{1}{4}$ and $x^{2}$ .

$3 x = \frac{x^{2}}{4} + 5$

Step 2

Subtract $\frac{x^{2}}{4}$ from both sides of the equation.

$3 x - \frac{x^{2}}{4} = 5$

Step 3

Subtract $5$ from both sides of the equation.

$3 x - \frac{x^{2}}{4} - 5 = 0$

Step 4

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

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Step 4.1

Apply the distributive property.

$4 (3 x) + 4 (- \frac{x^{2}}{4}) + 4 \cdot - 5 = 0$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $3$ by $4$ .

$12 x + 4 (- \frac{x^{2}}{4}) + 4 \cdot - 5 = 0$

Step 4.2.2

Cancel the common factor of $4$ .

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Step 4.2.2.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$12 x + 4 (\frac{- x^{2}}{4}) + 4 \cdot - 5 = 0$

Step 4.2.2.2

Cancel the common factor.

$12 x + 4 (\frac{- x^{2}}{4}) + 4 \cdot - 5 = 0$

Step 4.2.2.3

Rewrite the expression.

$12 x - x^{2} + 4 \cdot - 5 = 0$

Step 4.2.3

Multiply $4$ by $- 5$ .

$12 x - x^{2} - 20 = 0$

Step 4.3

Reorder $12 x$ and $- x^{2}$ .

$- x^{2} + 12 x - 20 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 1$ , $b = 12$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 1 \cdot - 20)}}{2 \cdot - 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 1 \cdot - 20}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 20$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot - 20}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 20$ .

$x = \frac{- 12 \pm \sqrt{144 - 80}}{2 \cdot - 1}$

Step 7.1.3

Subtract $80$ from $144$ .

$x = \frac{- 12 \pm \sqrt{64}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 12 \pm \sqrt{8^{2}}}{2 \cdot - 1}$

Step 7.1.5

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

$x = \frac{- 12 \pm 8}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12 \pm 8}{- 2}$

Step 7.3

Simplify $\frac{- 12 \pm 8}{- 2}$ .

$x = 6 \pm 4$

Step 8

The final answer is the combination of both solutions.

$x = 10, 2$