Finite Math Examples

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

Step 1

Simplify each term.

Tap for more steps...

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

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{1}{5})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{2 x}{5} + {(- \frac{1}{5})}^{2} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4

Simplify the equation.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Simplify each term.

Tap for more steps...

Step 4.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.1.1.1

Apply the product rule to $- \frac{1}{5}$ .

$x^{2} - \frac{2 x}{5} + {(- 1)}^{2} {(\frac{1}{5})}^{2} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.1.1.1.2

Apply the product rule to $\frac{1}{5}$ .

$x^{2} - \frac{2 x}{5} + {(- 1)}^{2} \frac{1^{2}}{5^{2}} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{2 x}{5} + 1 \frac{1^{2}}{5^{2}} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.1.1.3

Multiply $\frac{1^{2}}{5^{2}}$ by $1$ .

$x^{2} - \frac{2 x}{5} + \frac{1^{2}}{5^{2}} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.1.1.4

One to any power is one.

$x^{2} - \frac{2 x}{5} + \frac{1}{5^{2}} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.1.1.5

Raise $5$ to the power of $2$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + {(- \frac{1}{5})}^{2}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $\frac{4}{25} + {(- \frac{1}{5})}^{2}$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.2.1.1.1.1

Apply the product rule to $- \frac{1}{5}$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + {(- 1)}^{2} {(\frac{1}{5})}^{2}$

Step 4.2.1.1.1.2

Apply the product rule to $\frac{1}{5}$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + {(- 1)}^{2} \frac{1^{2}}{5^{2}}$

Step 4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + 1 \frac{1^{2}}{5^{2}}$

Step 4.2.1.1.3

Multiply $\frac{1^{2}}{5^{2}}$ by $1$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + \frac{1^{2}}{5^{2}}$

Step 4.2.1.1.4

One to any power is one.

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + \frac{1}{5^{2}}$

Step 4.2.1.1.5

Raise $5$ to the power of $2$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4}{25} + \frac{1}{25}$

Step 4.2.1.2

Simplify terms.

Tap for more steps...

Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{4 + 1}{25}$

Step 4.2.1.2.2

Add $4$ and $1$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{5}{25}$

Step 4.2.1.2.3

Cancel the common factor of $5$ and $25$ .

Tap for more steps...

Step 4.2.1.2.3.1

Factor $5$ out of $5$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{5 (1)}{25}$

Step 4.2.1.2.3.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1.2.3.2.1

Factor $5$ out of $25$ .

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{5 \cdot 1}{5 \cdot 5}$

Step 4.2.1.2.3.2.2

Cancel the common factor.

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{5 \cdot 1}{5 \cdot 5}$

Step 4.2.1.2.3.2.3

Rewrite the expression.

$x^{2} - \frac{2 x}{5} + \frac{1}{25} = \frac{1}{5}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{1}{5})}^{2}$ .

${(x - \frac{1}{5})}^{2} = \frac{1}{5}$

Step 6

Solve the equation for $x$ .

Tap for more steps...

Step 6.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 6.2

Simplify $\pm \sqrt{\frac{1}{5}}$ .

Tap for more steps...

Step 6.2.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

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

Step 6.2.2

Any root of $1$ is $1$ .

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

Step 6.2.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 6.2.4

Combine and simplify the denominator.

Tap for more steps...

Step 6.2.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 6.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 6.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 6.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.4.5

Add $1$ and $1$ .

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

Step 6.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 6.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x - \frac{1}{5} = \pm \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 6.2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 6.2.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.4.6.4.1

Cancel the common factor.

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

Step 6.2.4.6.4.2

Rewrite the expression.

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

Step 6.2.4.6.5

Evaluate the exponent.

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

Step 6.3

Add $\frac{1}{5}$ to both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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