Algebra Examples

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

Step 1

Set $4 x^{2} + x - \frac{1}{4}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Simplify.

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Step 2.1.2.1

Multiply $4$ by $4$ .

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

Step 2.1.2.2

Cancel the common factor of $4$ .

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Cancel the common factor.

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

Step 2.1.2.2.3

Rewrite the expression.

$16 x^{2} + 4 x - 1 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 16$ , $b = 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (16 \cdot - 1)}}{2 \cdot 16}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16 \cdot - 1}}{2 \cdot 16}$

Step 2.4.1.2

Multiply $- 4 \cdot 16 \cdot - 1$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64 \cdot - 1}}{2 \cdot 16}$

Step 2.4.1.2.2

Multiply $- 64$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 64}}{2 \cdot 16}$

Step 2.4.1.3

Add $16$ and $64$ .

$x = \frac{- 4 \pm \sqrt{80}}{2 \cdot 16}$

Step 2.4.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 2.4.1.4.1

Factor $16$ out of $80$ .

$x = \frac{- 4 \pm \sqrt{16 (5)}}{2 \cdot 16}$

Step 2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 16}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{5}}{2 \cdot 16}$

Step 2.4.2

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 \sqrt{5}}{32}$

Step 2.4.3

Simplify $\frac{- 4 \pm 4 \sqrt{5}}{32}$ .

$x = \frac{- 1 \pm \sqrt{5}}{8}$

Step 2.5

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{5}}{8}, - \frac{1 + \sqrt{5}}{8}$

$x = \frac{- 1 \pm \sqrt{5}}{8}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 1 \pm \sqrt{5}}{8}$

Decimal Form:

$x = 0.15450849 \dots, - 0.40450849 \dots$

Step 4