Algebra Examples

$\frac{x}{9} = \frac{x - 3}{x - 1}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$x (x - 1) = 9 (x - 3)$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify $x (x - 1)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + x (x - 1) = 9 (x - 3)$

Step 2.1.2

Simplify by multiplying through.

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Step 2.1.2.1

Apply the distributive property.

$x \cdot x + x \cdot - 1 = 9 (x - 3)$

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 = 9 (x - 3)$

Step 2.1.2.2.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x = 9 (x - 3)$

Step 2.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x = 9 (x - 3)$

Step 2.2

Simplify $9 (x - 3)$ .

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Step 2.2.1

Apply the distributive property.

$x^{2} - x = 9 x + 9 \cdot - 3$

Step 2.2.2

Multiply $9$ by $- 3$ .

$x^{2} - x = 9 x - 27$

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Subtract $9 x$ from both sides of the equation.

$x^{2} - x - 9 x = - 27$

Step 2.3.2

Subtract $9 x$ from $- x$ .

$x^{2} - 10 x = - 27$

Step 2.4

Add $27$ to both sides of the equation.

$x^{2} - 10 x + 27 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 1$ , $b = - 10$ , and $c = 27$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 27)}}{2 \cdot 1}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 27}}{2 \cdot 1}$

Step 2.7.1.2

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

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 27}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $27$ .

$x = \frac{10 \pm \sqrt{100 - 108}}{2 \cdot 1}$

Step 2.7.1.3

Subtract $108$ from $100$ .

$x = \frac{10 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.7.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.7.1.7.1

Factor $4$ out of $8$ .

$x = \frac{10 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.7.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.7.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{10 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 i \sqrt{2}}{2}$

Step 2.7.3

Simplify $\frac{10 \pm 2 i \sqrt{2}}{2}$ .

$x = 5 \pm i \sqrt{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = 5 + i \sqrt{2}, 5 - i \sqrt{2}$