Solve by Completing the Square 5n^2-2n-10=15

$5 n^{2} - 2 n - 10 = 15$

Step 1

Move all terms not containing $n$ to the right side of the equation.

Add $10$ to both sides of the equation.

$5 n^{2} - 2 n = 15 + 10$

Add $15$ and $10$ .

$5 n^{2} - 2 n = 25$

Step 2

Divide each term in $5 n^{2} - 2 n = 25$ by $5$ and simplify.

Divide each term in $5 n^{2} - 2 n = 25$ by $5$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $n^{2}$ by $1$ .

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

Move the negative in front of the fraction.

$n^{2} - \frac{2 n}{5} = \frac{25}{5}$

Simplify the right side.

Divide $25$ by $5$ .

$n^{2} - \frac{2 n}{5} = 5$

Step 3

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 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

To write $5$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

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

Combine $5$ and $\frac{25}{25}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $5$ by $25$ .

$n^{2} - \frac{2 n}{5} + \frac{1}{25} = \frac{125 + 1}{25}$

Add $125$ and $1$ .

$n^{2} - \frac{2 n}{5} + \frac{1}{25} = \frac{126}{25}$

Step 6

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

${(n - \frac{1}{5})}^{2} = \frac{126}{25}$

Step 7

Solve the equation for $n$ .

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

$n - \frac{1}{5} = \pm \sqrt{\frac{126}{25}}$

Simplify $\pm \sqrt{\frac{126}{25}}$ .

Rewrite $\sqrt{\frac{126}{25}}$ as $\frac{\sqrt{126}}{\sqrt{25}}$ .

$n - \frac{1}{5} = \pm \frac{\sqrt{126}}{\sqrt{25}}$

Simplify the numerator.

Rewrite $126$ as $3^{2} \cdot 14$ .

Factor $9$ out of $126$ .

$n - \frac{1}{5} = \pm \frac{\sqrt{9 (14)}}{\sqrt{25}}$

Rewrite $9$ as $3^{2}$ .

$n - \frac{1}{5} = \pm \frac{\sqrt{3^{2} \cdot 14}}{\sqrt{25}}$

Pull terms out from under the radical.

$n - \frac{1}{5} = \pm \frac{3 \sqrt{14}}{\sqrt{25}}$

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$n - \frac{1}{5} = \pm \frac{3 \sqrt{14}}{\sqrt{5^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$n - \frac{1}{5} = \pm \frac{3 \sqrt{14}}{5}$

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

$n = \pm \frac{3 \sqrt{14}}{5} + \frac{1}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = \pm \frac{3 \sqrt{14}}{5} + \frac{1}{5}$

Decimal Form:

$n = 2.44499443 \dots, - 2.04499443 \dots$

Solve by Completing the Square 5n^2-2n-10=15

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