Finite Math Examples

$x^{2} + {(x - 14)}^{2} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite ${(x - 14)}^{2}$ as $(x - 14) (x - 14)$ .

$x^{2} + (x - 14) (x - 14) = 0$

Step 1.2

Expand $(x - 14) (x - 14)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x^{2} + x (x - 14) - 14 (x - 14) = 0$

Step 1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 14 - 14 (x - 14) = 0$

Step 1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14 = 0$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14 = 0$

Step 1.3.1.2

Move $- 14$ to the left of $x$ .

$x^{2} + x^{2} - 14 x - 14 x - 14 \cdot - 14 = 0$

Step 1.3.1.3

Multiply $- 14$ by $- 14$ .

$x^{2} + x^{2} - 14 x - 14 x + 196 = 0$

Step 1.3.2

Subtract $14 x$ from $- 14 x$ .

$x^{2} + x^{2} - 28 x + 196 = 0$

Step 1.4

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 28 x + 196 = 0$

Step 1.5

Factor $2$ out of $2 x^{2} - 28 x + 196$ .

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Step 1.5.1

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

$2 (x^{2}) - 28 x + 196 = 0$

Step 1.5.2

Factor $2$ out of $- 28 x$ .

$2 (x^{2}) + 2 (- 14 x) + 196 = 0$

Step 1.5.3

Factor $2$ out of $196$ .

$2 x^{2} + 2 (- 14 x) + 2 \cdot 98 = 0$

Step 1.5.4

Factor $2$ out of $2 x^{2} + 2 (- 14 x)$ .

$2 (x^{2} - 14 x) + 2 \cdot 98 = 0$

Step 1.5.5

Factor $2$ out of $2 (x^{2} - 14 x) + 2 \cdot 98$ .

$2 (x^{2} - 14 x + 98) = 0$

Step 2

Divide each term in $2 (x^{2} - 14 x + 98) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (x^{2} - 14 x + 98) = 0$ by $2$ .

$\frac{2 (x^{2} - 14 x + 98)}{2} = \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 14 x + 98)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $x^{2} - 14 x + 98$ by $1$ .

$x^{2} - 14 x + 98 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x^{2} - 14 x + 98 = 0$

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 $a = 1$ , $b = - 14$ , and $c = 98$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 98)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 98}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 98$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 98}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $98$ .

$x = \frac{14 \pm \sqrt{196 - 392}}{2 \cdot 1}$

Step 5.1.3

Subtract $392$ from $196$ .

$x = \frac{14 \pm \sqrt{- 196}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{14 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{14 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Step 5.1.7

Rewrite $196$ as $14^{2}$ .

$x = \frac{14 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$x = \frac{14 \pm i \cdot 14}{2 \cdot 1}$

Step 5.1.9

Move $14$ to the left of $i$ .

$x = \frac{14 \pm 14 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{14 \pm 14 i}{2}$

Step 5.3

Simplify $\frac{14 \pm 14 i}{2}$ .

$x = 7 \pm 7 i$

Step 6

The final answer is the combination of both solutions.

$x = 7 + 7 i, 7 - 7 i$