Finite Math Examples

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

Step 1

Add $4$ to both sides of the equation.

${(n - 5)}^{2} + 4 = 0$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(n - 5)}^{2}$ as $(n - 5) (n - 5)$ .

$(n - 5) (n - 5) + 4 = 0$

Step 2.1.2

Expand $(n - 5) (n - 5)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$n (n - 5) - 5 (n - 5) + 4 = 0$

Step 2.1.2.2

Apply the distributive property.

$n \cdot n + n \cdot - 5 - 5 (n - 5) + 4 = 0$

Step 2.1.2.3

Apply the distributive property.

$n \cdot n + n \cdot - 5 - 5 n - 5 \cdot - 5 + 4 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $n$ by $n$ .

$n^{2} + n \cdot - 5 - 5 n - 5 \cdot - 5 + 4 = 0$

Step 2.1.3.1.2

Move $- 5$ to the left of $n$ .

$n^{2} - 5 \cdot n - 5 n - 5 \cdot - 5 + 4 = 0$

Step 2.1.3.1.3

Multiply $- 5$ by $- 5$ .

$n^{2} - 5 n - 5 n + 25 + 4 = 0$

Step 2.1.3.2

Subtract $5 n$ from $- 5 n$ .

$n^{2} - 10 n + 25 + 4 = 0$

Step 2.2

Add $25$ and $4$ .

$n^{2} - 10 n + 29 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 10$ , and $c = 29$ into the quadratic formula and solve for $n$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $29$ .

$n = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Step 5.1.3

Subtract $116$ from $100$ .

$n = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$n = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$n = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 5.1.7

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

$n = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$n = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Step 5.1.9

Move $4$ to the left of $i$ .

$n = \frac{10 \pm 4 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$n = \frac{10 \pm 4 i}{2}$

Step 5.3

Simplify $\frac{10 \pm 4 i}{2}$ .

$n = 5 \pm 2 i$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $29$ .

$n = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Step 6.1.3

Subtract $116$ from $100$ .

$n = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$n = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$n = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 6.1.7

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

$n = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 6.1.8

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

$n = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Step 6.1.9

Move $4$ to the left of $i$ .

$n = \frac{10 \pm 4 i}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$n = \frac{10 \pm 4 i}{2}$

Step 6.3

Simplify $\frac{10 \pm 4 i}{2}$ .

$n = 5 \pm 2 i$

Step 6.4

Change the $\pm$ to $+$ .

$n = 5 + 2 i$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $29$ .

$n = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Step 7.1.3

Subtract $116$ from $100$ .

$n = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$n = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$n = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 7.1.7

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

$n = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 7.1.8

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

$n = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Step 7.1.9

Move $4$ to the left of $i$ .

$n = \frac{10 \pm 4 i}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$n = \frac{10 \pm 4 i}{2}$

Step 7.3

Simplify $\frac{10 \pm 4 i}{2}$ .

$n = 5 \pm 2 i$

Step 7.4

Change the $\pm$ to $-$ .

$n = 5 - 2 i$

Step 8

The final answer is the combination of both solutions.

$n = 5 + 2 i, 5 - 2 i$