Algebra Examples

$x^{2} = \frac{8}{9} x$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Combine $\frac{8}{9}$ and $x$ .

$x^{2} = \frac{8 x}{9}$

Step 1.2

Subtract $\frac{8 x}{9}$ from both sides of the equation.

$x^{2} - \frac{8 x}{9} = 0$

Step 2

Multiply through by the least common denominator $9$ , then simplify.

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Step 2.1

Apply the distributive property.

$9 x^{2} + 9 (- \frac{8 x}{9}) = 0$

Step 2.2

Cancel the common factor of $9$ .

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Step 2.2.1

Move the leading negative in $- \frac{8 x}{9}$ into the numerator.

$9 x^{2} + 9 (\frac{- 8 x}{9}) = 0$

Step 2.2.2

Cancel the common factor.

$9 x^{2} + 9 (\frac{- 8 x}{9}) = 0$

Step 2.2.3

Rewrite the expression.

$9 x^{2} - 8 x = 0$

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 $a = 9$ , $b = - 8$ , and $c = 0$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (9 \cdot 0)}}{2 \cdot 9}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 9 \cdot 0}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 0$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{8 \pm \sqrt{64 - 36 \cdot 0}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $0$ .

$x = \frac{8 \pm \sqrt{64 + 0}}{2 \cdot 9}$

Step 5.1.3

Add $64$ and $0$ .

$x = \frac{8 \pm \sqrt{64}}{2 \cdot 9}$

Step 5.1.4

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

$x = \frac{8 \pm \sqrt{8^{2}}}{2 \cdot 9}$

Step 5.1.5

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

$x = \frac{8 \pm 8}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{8 \pm 8}{18}$

Step 5.3

Simplify $\frac{8 \pm 8}{18}$ .

$x = \frac{4 \pm 4}{9}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{8}{9}, 0$