Algebra Examples

$4 x^{2} - 27 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 4$ , $b = 0$ , and $c = - 27$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (4 \cdot - 27)}}{2 \cdot 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 4 \cdot - 27}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 4 \cdot 4 \cdot - 27$ .

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Step 3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{0 \pm \sqrt{0 - 16 \cdot - 27}}{2 \cdot 4}$

Step 3.1.2.2

Multiply $- 16$ by $- 27$ .

$x = \frac{0 \pm \sqrt{0 + 432}}{2 \cdot 4}$

Step 3.1.3

Add $0$ and $432$ .

$x = \frac{0 \pm \sqrt{432}}{2 \cdot 4}$

Step 3.1.4

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 3.1.4.1

Factor $144$ out of $432$ .

$x = \frac{0 \pm \sqrt{144 (3)}}{2 \cdot 4}$

Step 3.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{0 \pm \sqrt{12^{2} \cdot 3}}{2 \cdot 4}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{0 \pm 12 \sqrt{3}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$x = \frac{0 \pm 12 \sqrt{3}}{8}$

Step 3.3

Simplify $\frac{0 \pm 12 \sqrt{3}}{8}$ .

$x = \frac{\pm 3 \sqrt{3}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\pm 3 \sqrt{3}}{2}$

Decimal Form:

$x = 2.59807621 \dots, - 2.59807621 \dots$