Finite Math Examples

$x^{2} + {(x + 2)}^{2} = {(x + 4)}^{2} - 65$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract ${(x + 4)}^{2}$ from both sides of the equation.

$x^{2} + {(x + 2)}^{2} - {(x + 4)}^{2} = - 65$

Step 1.2

Add $65$ to both sides of the equation.

$x^{2} + {(x + 2)}^{2} - {(x + 4)}^{2} + 65 = 0$

Step 2

Simplify $x^{2} + {(x + 2)}^{2} - {(x + 4)}^{2} + 65$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$x^{2} + (x + 2) (x + 2) - {(x + 4)}^{2} + 65 = 0$

Step 2.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x^{2} + x (x + 2) + 2 (x + 2) - {(x + 4)}^{2} + 65 = 0$

Step 2.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 2 + 2 (x + 2) - {(x + 4)}^{2} + 65 = 0$

Step 2.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - {(x + 4)}^{2} + 65 = 0$

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 $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - {(x + 4)}^{2} + 65 = 0$

Step 2.1.3.1.2

Move $2$ to the left of $x$ .

$x^{2} + x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - {(x + 4)}^{2} + 65 = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$x^{2} + x^{2} + 2 x + 2 x + 4 - {(x + 4)}^{2} + 65 = 0$

Step 2.1.3.2

Add $2 x$ and $2 x$ .

$x^{2} + x^{2} + 4 x + 4 - {(x + 4)}^{2} + 65 = 0$

Step 2.1.4

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$x^{2} + x^{2} + 4 x + 4 - ((x + 4) (x + 4)) + 65 = 0$

Step 2.1.5

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$x^{2} + x^{2} + 4 x + 4 - (x (x + 4) + 4 (x + 4)) + 65 = 0$

Step 2.1.5.2

Apply the distributive property.

$x^{2} + x^{2} + 4 x + 4 - (x \cdot x + x \cdot 4 + 4 (x + 4)) + 65 = 0$

Step 2.1.5.3

Apply the distributive property.

$x^{2} + x^{2} + 4 x + 4 - (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 65 = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + 4 x + 4 - (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) + 65 = 0$

Step 2.1.6.1.2

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

$x^{2} + x^{2} + 4 x + 4 - (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) + 65 = 0$

Step 2.1.6.1.3

Multiply $4$ by $4$ .

$x^{2} + x^{2} + 4 x + 4 - (x^{2} + 4 x + 4 x + 16) + 65 = 0$

Step 2.1.6.2

Add $4 x$ and $4 x$ .

$x^{2} + x^{2} + 4 x + 4 - (x^{2} + 8 x + 16) + 65 = 0$

Step 2.1.7

Apply the distributive property.

$x^{2} + x^{2} + 4 x + 4 - x^{2} - (8 x) - 1 \cdot 16 + 65 = 0$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $8$ by $- 1$ .

$x^{2} + x^{2} + 4 x + 4 - x^{2} - 8 x - 1 \cdot 16 + 65 = 0$

Step 2.1.8.2

Multiply $- 1$ by $16$ .

$x^{2} + x^{2} + 4 x + 4 - x^{2} - 8 x - 16 + 65 = 0$

Step 2.2

Combine the opposite terms in $x^{2} + x^{2} + 4 x + 4 - x^{2} - 8 x - 16 + 65$ .

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Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$x^{2} + 4 x + 4 + 0 - 8 x - 16 + 65 = 0$

Step 2.2.2

Add $x^{2} + 4 x + 4$ and $0$ .

$x^{2} + 4 x + 4 - 8 x - 16 + 65 = 0$

Step 2.3

Subtract $8 x$ from $4 x$ .

$x^{2} - 4 x + 4 - 16 + 65 = 0$

Step 2.4

Subtract $16$ from $4$ .

$x^{2} - 4 x - 12 + 65 = 0$

Step 2.5

Add $- 12$ and $65$ .

$x^{2} - 4 x + 53 = 0$

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 $a = 1$ , $b = - 4$ , and $c = 53$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 53)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 53}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{4 \pm \sqrt{16 - 212}}{2 \cdot 1}$

Step 5.1.3

Subtract $212$ from $16$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Step 5.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$x = \frac{4 \pm i \cdot 14}{2 \cdot 1}$

Step 5.1.9

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

$x = \frac{4 \pm 14 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 14 i}{2}$

Step 5.3

Simplify $\frac{4 \pm 14 i}{2}$ .

$x = 2 \pm 7 i$

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 53}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{4 \pm \sqrt{16 - 212}}{2 \cdot 1}$

Step 6.1.3

Subtract $212$ from $16$ .

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Step 6.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

Step 6.1.8

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

$x = \frac{4 \pm i \cdot 14}{2 \cdot 1}$

Step 6.1.9

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

$x = \frac{4 \pm 14 i}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 14 i}{2}$

Step 6.3

Simplify $\frac{4 \pm 14 i}{2}$ .

$x = 2 \pm 7 i$

Step 6.4

Change the $\pm$ to $+$ .

$x = 2 + 7 i$

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 53}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{4 \pm \sqrt{16 - 212}}{2 \cdot 1}$

Step 7.1.3

Subtract $212$ from $16$ .

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

Step 7.1.4

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

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

Step 7.1.5

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

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

Step 7.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Step 7.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

Step 7.1.8

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

$x = \frac{4 \pm i \cdot 14}{2 \cdot 1}$

Step 7.1.9

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

$x = \frac{4 \pm 14 i}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 14 i}{2}$

Step 7.3

Simplify $\frac{4 \pm 14 i}{2}$ .

$x = 2 \pm 7 i$

Step 7.4

Change the $\pm$ to $-$ .

$x = 2 - 7 i$

Step 8

The final answer is the combination of both solutions.

$x = 2 + 7 i, 2 - 7 i$