Finite Math Examples

$x^{2} - \frac{2}{5} x = \frac{4}{25}$

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{2}{5}$ .

$x^{2} - \frac{x \cdot 2}{5} = \frac{4}{25}$

Step 1.2

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

$x^{2} - \frac{2 x}{5} = \frac{4}{25}$

Step 2

Subtract $\frac{4}{25}$ from both sides of the equation.

$x^{2} - \frac{2 x}{5} - \frac{4}{25} = 0$

Step 3

Multiply through by the least common denominator $25$ , then simplify.

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Step 3.1

Apply the distributive property.

$25 x^{2} + 25 (- \frac{2 x}{5}) + 25 (- \frac{4}{25}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{2 x}{5}$ into the numerator.

$25 x^{2} + 25 (\frac{- 2 x}{5}) + 25 (- \frac{4}{25}) = 0$

Step 3.2.1.2

Factor $5$ out of $25$ .

$25 x^{2} + 5 (5) (\frac{- 2 x}{5}) + 25 (- \frac{4}{25}) = 0$

Step 3.2.1.3

Cancel the common factor.

$25 x^{2} + 5 \cdot (5 (\frac{- 2 x}{5})) + 25 (- \frac{4}{25}) = 0$

Step 3.2.1.4

Rewrite the expression.

$25 x^{2} + 5 (- 2 x) + 25 (- \frac{4}{25}) = 0$

Step 3.2.2

Multiply $- 2$ by $5$ .

$25 x^{2} - 10 x + 25 (- \frac{4}{25}) = 0$

Step 3.2.3

Cancel the common factor of $25$ .

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Step 3.2.3.1

Move the leading negative in $- \frac{4}{25}$ into the numerator.

$25 x^{2} - 10 x + 25 (\frac{- 4}{25}) = 0$

Step 3.2.3.2

Cancel the common factor.

$25 x^{2} - 10 x + 25 (\frac{- 4}{25}) = 0$

Step 3.2.3.3

Rewrite the expression.

$25 x^{2} - 10 x - 4 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 25$ , $b = - 10$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (25 \cdot - 4)}}{2 \cdot 25}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 25 \cdot - 4}}{2 \cdot 25}$

Step 6.1.2

Multiply $- 4 \cdot 25 \cdot - 4$ .

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Step 6.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{10 \pm \sqrt{100 - 100 \cdot - 4}}{2 \cdot 25}$

Step 6.1.2.2

Multiply $- 100$ by $- 4$ .

$x = \frac{10 \pm \sqrt{100 + 400}}{2 \cdot 25}$

Step 6.1.3

Add $100$ and $400$ .

$x = \frac{10 \pm \sqrt{500}}{2 \cdot 25}$

Step 6.1.4

Rewrite $500$ as $10^{2} \cdot 5$ .

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Step 6.1.4.1

Factor $100$ out of $500$ .

$x = \frac{10 \pm \sqrt{100 (5)}}{2 \cdot 25}$

Step 6.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{10 \pm \sqrt{10^{2} \cdot 5}}{2 \cdot 25}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 10 \sqrt{5}}{2 \cdot 25}$

Step 6.2

Multiply $2$ by $25$ .

$x = \frac{10 \pm 10 \sqrt{5}}{50}$

Step 6.3

Simplify $\frac{10 \pm 10 \sqrt{5}}{50}$ .

$x = \frac{1 \pm \sqrt{5}}{5}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{5}}{5}, \frac{1 - \sqrt{5}}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{5}}{5}, \frac{1 - \sqrt{5}}{5}$

Decimal Form:

$x = 0.64721359 \dots, - 0.24721359 \dots$