Algebra Examples

x^{2} = 27

$x^{2} = 27$

Step 1

Subtract

27

$27$ from both sides of the equation.

x^{2} - 27 = 0

$x^{2} - 27 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

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

Step 3

Substitute the values

a = 1

$a = 1$ ,

b = 0

$b = 0$ , and

c = - 27

$c = - 27$ into the quadratic formula and solve for

x

$x$ .

\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot - 27)}}{2 \cdot 1}

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot - 27)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot - 27}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot - 27}}{2 \cdot 1}$

Step 4.1.2

Multiply

- 4 \cdot 1 \cdot - 27

$- 4 \cdot 1 \cdot - 27$ .

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{0 \pm \sqrt{0 - 4 \cdot - 27}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{0 - 4 \cdot - 27}}{2 \cdot 1}$

Step 4.1.2.2

Multiply

- 4

$- 4$ by

- 27

$- 27$ .

x = \frac{0 \pm \sqrt{0 + 108}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{0 + 108}}{2 \cdot 1}$

x = \frac{0 \pm \sqrt{0 + 108}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{0 + 108}}{2 \cdot 1}$

Step 4.1.3

Add

0

$0$ and

108

$108$ .

x = \frac{0 \pm \sqrt{108}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{108}}{2 \cdot 1}$

Step 4.1.4

Rewrite

108

$108$ as

6^{2} \cdot 3

$6^{2} \cdot 3$ .

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Step 4.1.4.1

Factor

36

$36$ out of

108

$108$ .

x = \frac{0 \pm \sqrt{36 (3)}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite

36

$36$ as

6^{2}

$6^{2}$ .

x = \frac{0 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

x = \frac{0 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{0 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

x = \frac{0 \pm 6 \sqrt{3}}{2 \cdot 1}

$x = \frac{0 \pm 6 \sqrt{3}}{2 \cdot 1}$

x = \frac{0 \pm 6 \sqrt{3}}{2 \cdot 1}

$x = \frac{0 \pm 6 \sqrt{3}}{2 \cdot 1}$

Step 4.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{0 \pm 6 \sqrt{3}}{2}

$x = \frac{0 \pm 6 \sqrt{3}}{2}$

Step 4.3

Simplify

\frac{0 \pm 6 \sqrt{3}}{2}

$\frac{0 \pm 6 \sqrt{3}}{2}$ .

x = \pm 3 \sqrt{3}

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

x = \pm 3 \sqrt{3}

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

Step 5

The result can be shown in multiple forms.

Exact Form:

x = \pm 3 \sqrt{3}

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

Decimal Form:

x = 5.19615242 \dots, - 5.19615242 \dots

$x = 5.19615242 \dots, - 5.19615242 \dots$