Algebra Examples

$0.2 = \frac{x^{2}}{45 - x}$

Step 1

Subtract $\frac{x^{2}}{45 - x}$ from both sides of the equation.

$0.2 - \frac{x^{2}}{45 - x} = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 45 - x, 1$

Step 2.2

Remove parentheses.

$1, 45 - x, 1$

Step 2.3

The LCM of one and any expression is the expression.

$45 - x$

Step 3

Multiply each term in $0.2 - \frac{x^{2}}{45 - x} = 0$ by $45 - x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $0.2 - \frac{x^{2}}{45 - x} = 0$ by $45 - x$ .

$0.2 (45 - x) - \frac{x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$0.2 \cdot 45 + 0.2 (- x) - \frac{x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2.1.2

Multiply $0.2$ by $45$ .

$9 + 0.2 (- x) - \frac{x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2.1.3

Multiply $- 1$ by $0.2$ .

$9 - 0.2 x - \frac{x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2.1.4

Cancel the common factor of $45 - x$ .

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Step 3.2.1.4.1

Move the leading negative in $- \frac{x^{2}}{45 - x}$ into the numerator.

$9 - 0.2 x + \frac{- x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2.1.4.2

Cancel the common factor.

$9 - 0.2 x + \frac{- x^{2}}{45 - x} (45 - x) = 0 (45 - x)$

Step 3.2.1.4.3

Rewrite the expression.

$9 - 0.2 x - x^{2} = 0 (45 - x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $45 - x$ .

$9 - 0.2 x - x^{2} = 0$

Step 4

Solve the equation.

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Step 4.1

Factor $- 0.2$ out of $9 - 0.2 x - x^{2}$ .

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Step 4.1.1

Reorder the expression.

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Step 4.1.1.1

Move $9$ .

$- 0.2 x - x^{2} + 9 = 0$

Step 4.1.1.2

Reorder $- 0.2 x$ and $- x^{2}$ .

$- x^{2} - 0.2 x + 9 = 0$

Step 4.1.2

Factor $- 0.2$ out of $- x^{2}$ .

$- 0.2 (5 x^{2}) - 0.2 x + 9 = 0$

Step 4.1.3

Factor $- 0.2$ out of $- 0.2 x$ .

$- 0.2 (5 x^{2}) - 0.2 x + 9 = 0$

Step 4.1.4

Factor $- 0.2$ out of $9$ .

$- 0.2 (5 x^{2}) - 0.2 x - 0.2 \cdot - 45 = 0$

Step 4.1.5

Factor $- 0.2$ out of $- 0.2 (5 x^{2}) - 0.2 (x)$ .

$- 0.2 (5 x^{2} + x) - 0.2 \cdot - 45 = 0$

Step 4.1.6

Factor $- 0.2$ out of $- 0.2 (5 x^{2} + x) - 0.2 (- 45)$ .

$- 0.2 (5 x^{2} + x - 45) = 0$

Step 4.2

Divide each term in $- 0.2 (5 x^{2} + x - 45) = 0$ by $- 0.2$ and simplify.

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Step 4.2.1

Divide each term in $- 0.2 (5 x^{2} + x - 45) = 0$ by $- 0.2$ .

$\frac{- 0.2 (5 x^{2} + x - 45)}{- 0.2} = \frac{0}{- 0.2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 0.2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 0.2 (5 x^{2} + x - 45)}{- 0.2} = \frac{0}{- 0.2}$

Step 4.2.2.1.2

Divide $5 x^{2} + x - 45$ by $1$ .

$5 x^{2} + x - 45 = \frac{0}{- 0.2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $0$ by $- 0.2$ .

$5 x^{2} + x - 45 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 5$ , $b = 1$ , and $c = - 45$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (5 \cdot - 45)}}{2 \cdot 5}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 45}}{2 \cdot 5}$

Step 4.5.1.2

Multiply $- 4 \cdot 5 \cdot - 45$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 45}}{2 \cdot 5}$

Step 4.5.1.2.2

Multiply $- 20$ by $- 45$ .

$x = \frac{- 1 \pm \sqrt{1 + 900}}{2 \cdot 5}$

Step 4.5.1.3

Add $1$ and $900$ .

$x = \frac{- 1 \pm \sqrt{901}}{2 \cdot 5}$

Step 4.5.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{901}}{10}$

Step 4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 45}}{2 \cdot 5}$

Step 4.6.1.2

Multiply $- 4 \cdot 5 \cdot - 45$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 45}}{2 \cdot 5}$

Step 4.6.1.2.2

Multiply $- 20$ by $- 45$ .

$x = \frac{- 1 \pm \sqrt{1 + 900}}{2 \cdot 5}$

Step 4.6.1.3

Add $1$ and $900$ .

$x = \frac{- 1 \pm \sqrt{901}}{2 \cdot 5}$

Step 4.6.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{901}}{10}$

Step 4.6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{901}}{10}$

Step 4.6.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{901}}{10}$

Step 4.6.5

Factor $- 1$ out of $\sqrt{901}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{901})}{10}$

Step 4.6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{901})$ .

$x = \frac{- 1 (1 - \sqrt{901})}{10}$

Step 4.6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{901}}{10}$

Step 4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 45}}{2 \cdot 5}$

Step 4.7.1.2

Multiply $- 4 \cdot 5 \cdot - 45$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 45}}{2 \cdot 5}$

Step 4.7.1.2.2

Multiply $- 20$ by $- 45$ .

$x = \frac{- 1 \pm \sqrt{1 + 900}}{2 \cdot 5}$

Step 4.7.1.3

Add $1$ and $900$ .

$x = \frac{- 1 \pm \sqrt{901}}{2 \cdot 5}$

Step 4.7.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{901}}{10}$

Step 4.7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{901}}{10}$

Step 4.7.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{901}}{10}$

Step 4.7.5

Factor $- 1$ out of $- \sqrt{901}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{901})}{10}$

Step 4.7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{901})$ .

$x = \frac{- 1 (1 + \sqrt{901})}{10}$

Step 4.7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{901}}{10}$

Step 4.8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{901}}{10}, - \frac{1 + \sqrt{901}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{901}}{10}, - \frac{1 + \sqrt{901}}{10}$

Decimal Form:

$x = 2.90166620 \dots, - 3.10166620 \dots$