Algebra Examples

$0.2 = \frac{x^{2}}{65 - x}$

Step 1

Rewrite the equation as $\frac{x^{2}}{65 - x} = 0.2$ .

$\frac{x^{2}}{65 - x} = 0.2$

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.

$65 - x, 1$

Step 2.2

Remove parentheses.

$65 - x, 1$

Step 2.3

The LCM of one and any expression is the expression.

$65 - x$

Step 3

Multiply each term in $\frac{x^{2}}{65 - x} = 0.2$ by $65 - x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{x^{2}}{65 - x} = 0.2$ by $65 - x$ .

$\frac{x^{2}}{65 - x} (65 - x) = 0.2 (65 - x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $65 - x$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{x^{2}}{65 - x} (65 - x) = 0.2 (65 - x)$

Step 3.2.1.2

Rewrite the expression.

$x^{2} = 0.2 (65 - x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$x^{2} = 0.2 \cdot 65 + 0.2 (- x)$

Step 3.3.2

Multiply.

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Step 3.3.2.1

Multiply $0.2$ by $65$ .

$x^{2} = 13 + 0.2 (- x)$

Step 3.3.2.2

Multiply $- 1$ by $0.2$ .

$x^{2} = 13 - 0.2 x$

Step 4

Solve the equation.

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Step 4.1

Add $0.2 x$ to both sides of the equation.

$x^{2} + 0.2 x = 13$

Step 4.2

Subtract $13$ from both sides of the equation.

$x^{2} + 0.2 x - 13 = 0$

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 $a = 1$ , $b = 0.2$ , and $c = - 13$ into the quadratic formula and solve for $x$ .

$\frac{- 0.2 \pm \sqrt{{0.2}^{2} - 4 \cdot (1 \cdot - 13)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $0.2$ to the power of $2$ .

$x = \frac{- 0.2 \pm \sqrt{0.04 - 4 \cdot 1 \cdot - 13}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 13$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 0.2 \pm \sqrt{0.04 - 4 \cdot - 13}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 13$ .

$x = \frac{- 0.2 \pm \sqrt{0.04 + 52}}{2 \cdot 1}$

Step 4.5.1.3

Add $0.04$ and $52$ .

$x = \frac{- 0.2 \pm \sqrt{52.04}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 0.2 \pm \sqrt{52.04}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = 3.50693775, - 3.70693775$