Finite Math Examples

$\frac{x^{2}}{2} + 4 x + 10 = 0$

Step 1

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2}}{2} + 4 x \cdot \frac{2}{2} + 10 = 0$

Step 2

Combine $4 x$ and $\frac{2}{2}$ .

$\frac{x^{2}}{2} + \frac{4 x \cdot 2}{2} + 10 = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{x^{2} + 4 x \cdot 2}{2} + 10 = 0$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $x^{2} + 4 x \cdot 2$ .

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Step 4.1.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 4 x \cdot 2}{2} + 10 = 0$

Step 4.1.2

Factor $x$ out of $4 x \cdot 2$ .

$\frac{x \cdot x + x (4 \cdot 2)}{2} + 10 = 0$

Step 4.1.3

Factor $x$ out of $x \cdot x + x (4 \cdot 2)$ .

$\frac{x (x + 4 \cdot 2)}{2} + 10 = 0$

Step 4.2

Multiply $4$ by $2$ .

$\frac{x (x + 8)}{2} + 10 = 0$

Step 5

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x (x + 8)}{2} + 10 \cdot \frac{2}{2} = 0$

Step 6

Combine $10$ and $\frac{2}{2}$ .

$\frac{x (x + 8)}{2} + \frac{10 \cdot 2}{2} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{x (x + 8) + 10 \cdot 2}{2} = 0$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 8 + 10 \cdot 2}{2} = 0$

Step 8.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 8 + 10 \cdot 2}{2} = 0$

Step 8.3

Move $8$ to the left of $x$ .

$\frac{x^{2} + 8 x + 10 \cdot 2}{2} = 0$

Step 8.4

Multiply $10$ by $2$ .

$\frac{x^{2} + 8 x + 20}{2} = 0$

Step 9

Set the numerator equal to zero.

$x^{2} + 8 x + 20 = 0$

Step 10

Solve the equation for $x$ .

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Step 10.1

Use the quadratic formula to find the solutions.

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

Step 10.2

Substitute the values $a = 1$ , $b = 8$ , and $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Step 10.3

Simplify.

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Step 10.3.1

Simplify the numerator.

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Step 10.3.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 10.3.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 10.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 20}}{2 \cdot 1}$

Step 10.3.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{- 8 \pm \sqrt{64 - 80}}{2 \cdot 1}$

Step 10.3.1.3

Subtract $80$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 10.3.1.4

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

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 10.3.1.5

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

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.3.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.3.1.7

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

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 10.3.1.8

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

$x = \frac{- 8 \pm i \cdot 4}{2 \cdot 1}$

Step 10.3.1.9

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

$x = \frac{- 8 \pm 4 i}{2 \cdot 1}$

Step 10.3.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 4 i}{2}$

Step 10.3.3

Simplify $\frac{- 8 \pm 4 i}{2}$ .

$x = - 4 \pm 2 i$

Step 10.4

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

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Step 10.4.1

Simplify the numerator.

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Step 10.4.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 10.4.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 10.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 20}}{2 \cdot 1}$

Step 10.4.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{- 8 \pm \sqrt{64 - 80}}{2 \cdot 1}$

Step 10.4.1.3

Subtract $80$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 10.4.1.4

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

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 10.4.1.5

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

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.4.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.4.1.7

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

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 10.4.1.8

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

$x = \frac{- 8 \pm i \cdot 4}{2 \cdot 1}$

Step 10.4.1.9

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

$x = \frac{- 8 \pm 4 i}{2 \cdot 1}$

Step 10.4.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 4 i}{2}$

Step 10.4.3

Simplify $\frac{- 8 \pm 4 i}{2}$ .

$x = - 4 \pm 2 i$

Step 10.4.4

Change the $\pm$ to $+$ .

$x = - 4 + 2 i$

Step 10.5

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

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Step 10.5.1

Simplify the numerator.

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Step 10.5.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 10.5.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 10.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 20}}{2 \cdot 1}$

Step 10.5.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{- 8 \pm \sqrt{64 - 80}}{2 \cdot 1}$

Step 10.5.1.3

Subtract $80$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 10.5.1.4

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

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 10.5.1.5

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

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.5.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 10.5.1.7

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

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 10.5.1.8

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

$x = \frac{- 8 \pm i \cdot 4}{2 \cdot 1}$

Step 10.5.1.9

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

$x = \frac{- 8 \pm 4 i}{2 \cdot 1}$

Step 10.5.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 4 i}{2}$

Step 10.5.3

Simplify $\frac{- 8 \pm 4 i}{2}$ .

$x = - 4 \pm 2 i$

Step 10.5.4

Change the $\pm$ to $-$ .

$x = - 4 - 2 i$

Step 10.6

The final answer is the combination of both solutions.

$x = - 4 + 2 i, - 4 - 2 i$