Solve by Completing the Square 2n^2-5n-4=6

$2 n^{2} - 5 n - 4 = 6$

Step 1

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

Add $4$ to both sides of the equation.

$2 n^{2} - 5 n = 6 + 4$

Add $6$ and $4$ .

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

Step 2

Divide each term in $2 n^{2} - 5 n = 10$ by $2$ and simplify.

Divide each term in $2 n^{2} - 5 n = 10$ by $2$ .

$\frac{2 n^{2}}{2} + \frac{- 5 n}{2} = \frac{10}{2}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n^{2}}{2} + \frac{- 5 n}{2} = \frac{10}{2}$

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

$n^{2} + \frac{- 5 n}{2} = \frac{10}{2}$

Move the negative in front of the fraction.

$n^{2} - \frac{5 n}{2} = \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{5}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$n^{2} - \frac{5 n}{2} + {(- \frac{5}{4})}^{2} = 5 + {(- \frac{5}{4})}^{2}$

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 $- \frac{5}{4}$ .

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

Apply the product rule to $\frac{5}{4}$ .

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

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

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

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

$n^{2} - \frac{5 n}{2} + \frac{5^{2}}{4^{2}} = 5 + {(- \frac{5}{4})}^{2}$

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

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

Raise $4$ to the power of $2$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + {(- \frac{5}{4})}^{2}$

Simplify the right side.

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

Simplify each term.

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

Apply the product rule to $- \frac{5}{4}$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + {(- 1)}^{2} {(\frac{5}{4})}^{2}$

Apply the product rule to $\frac{5}{4}$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + {(- 1)}^{2} \frac{5^{2}}{4^{2}}$

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

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + 1 \frac{5^{2}}{4^{2}}$

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + \frac{5^{2}}{4^{2}}$

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

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + \frac{25}{4^{2}}$

Raise $4$ to the power of $2$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 + \frac{25}{16}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = 5 \cdot \frac{16}{16} + \frac{25}{16}$

Combine $5$ and $\frac{16}{16}$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = \frac{5 \cdot 16}{16} + \frac{25}{16}$

Combine the numerators over the common denominator.

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = \frac{5 \cdot 16 + 25}{16}$

Simplify the numerator.

Multiply $5$ by $16$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = \frac{80 + 25}{16}$

Add $80$ and $25$ .

$n^{2} - \frac{5 n}{2} + \frac{25}{16} = \frac{105}{16}$

Step 6

Factor the perfect trinomial square into ${(n - \frac{5}{4})}^{2}$ .

${(n - \frac{5}{4})}^{2} = \frac{105}{16}$

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.

$n - \frac{5}{4} = \pm \sqrt{\frac{105}{16}}$

Simplify $\pm \sqrt{\frac{105}{16}}$ .

Rewrite $\sqrt{\frac{105}{16}}$ as $\frac{\sqrt{105}}{\sqrt{16}}$ .

$n - \frac{5}{4} = \pm \frac{\sqrt{105}}{\sqrt{16}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$n - \frac{5}{4} = \pm \frac{\sqrt{105}}{\sqrt{4^{2}}}$

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

$n - \frac{5}{4} = \pm \frac{\sqrt{105}}{4}$

Add $\frac{5}{4}$ to both sides of the equation.

$n = \pm \frac{\sqrt{105}}{4} + \frac{5}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = \pm \frac{\sqrt{105}}{4} + \frac{5}{4}$

Decimal Form:

$n = 3.81173769 \dots, - 1.31173769 \dots$

Solve by Completing the Square 2n^2-5n-4=6

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