Algebra Examples

$\frac{1}{18} x^{2} - \frac{1}{9} x = 1$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Combine $\frac{1}{18}$ and $x^{2}$ .

$\frac{x^{2}}{18} - \frac{1}{9} x = 1$

Step 1.1.1.2

Combine $x$ and $\frac{1}{9}$ .

$\frac{x^{2}}{18} - \frac{x}{9} = 1$

Step 1.2

Subtract $1$ from both sides of the equation.

$\frac{x^{2}}{18} - \frac{x}{9} - 1 = 0$

Step 2

Multiply through by the least common denominator $18$ , then simplify.

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Step 2.1

Apply the distributive property.

$18 (\frac{x^{2}}{18}) + 18 (- \frac{x}{9}) + 18 \cdot - 1 = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $18$ .

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Step 2.2.1.1

Cancel the common factor.

$18 (\frac{x^{2}}{18}) + 18 (- \frac{x}{9}) + 18 \cdot - 1 = 0$

Step 2.2.1.2

Rewrite the expression.

$x^{2} + 18 (- \frac{x}{9}) + 18 \cdot - 1 = 0$

Step 2.2.2

Cancel the common factor of $9$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$x^{2} + 18 (\frac{- x}{9}) + 18 \cdot - 1 = 0$

Step 2.2.2.2

Factor $9$ out of $18$ .

$x^{2} + 9 (2) (\frac{- x}{9}) + 18 \cdot - 1 = 0$

Step 2.2.2.3

Cancel the common factor.

$x^{2} + 9 \cdot (2 (\frac{- x}{9})) + 18 \cdot - 1 = 0$

Step 2.2.2.4

Rewrite the expression.

$x^{2} + 2 (- x) + 18 \cdot - 1 = 0$

Step 2.2.3

Multiply $- 1$ by $2$ .

$x^{2} - 2 x + 18 \cdot - 1 = 0$

Step 2.2.4

Multiply $18$ by $- 1$ .

$x^{2} - 2 x - 18 = 0$

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 $a = 1$ , $b = - 2$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 18)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 18}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 1}$

Step 5.1.3

Add $4$ and $72$ .

$x = \frac{2 \pm \sqrt{76}}{2 \cdot 1}$

Step 5.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 5.1.4.1

Factor $4$ out of $76$ .

$x = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{19}}{2}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{19}}{2}$ .

$x = 1 \pm \sqrt{19}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 1 \pm \sqrt{19}$

Decimal Form:

$x = 5.35889894 \dots, - 3.35889894 \dots$