Finite Math Examples

${(x + 3)}^{2} + {(x - 3)}^{2} = 0$

Step 1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(x + 3) (x + 3) + {(x - 3)}^{2} = 0$

Step 2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + 3) + 3 (x + 3) + {(x - 3)}^{2} = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3) + {(x - 3)}^{2} = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {(x - 3)}^{2} = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + {(x - 3)}^{2} = 0$

Step 3.1.2

Move $3$ to the left of $x$ .

$x^{2} + 3 x + 3 x + 3 \cdot 3 + {(x - 3)}^{2} = 0$

Step 3.1.3

Multiply $3$ by $3$ .

$x^{2} + 3 x + 3 x + 9 + {(x - 3)}^{2} = 0$

Step 3.2

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9 + {(x - 3)}^{2} = 0$

Step 4

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$x^{2} + 6 x + 9 + (x - 3) (x - 3) = 0$

Step 5

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$x^{2} + 6 x + 9 + x (x - 3) - 3 (x - 3) = 0$

Step 5.2

Apply the distributive property.

$x^{2} + 6 x + 9 + x \cdot x + x \cdot - 3 - 3 (x - 3) = 0$

Step 5.3

Apply the distributive property.

$x^{2} + 6 x + 9 + x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 = 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

$x^{2} + 6 x + 9 + x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 = 0$

Step 6.1.2

Move $- 3$ to the left of $x$ .

$x^{2} + 6 x + 9 + x^{2} - 3 x - 3 x - 3 \cdot - 3 = 0$

Step 6.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} + 6 x + 9 + x^{2} - 3 x - 3 x + 9 = 0$

Step 6.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} + 6 x + 9 + x^{2} - 6 x + 9 = 0$

Step 7

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

$2 x^{2} + 6 x + 9 - 6 x + 9 = 0$

Step 8

Subtract $6 x$ from $6 x$ .

$2 x^{2} + 0 + 9 + 9 = 0$

Step 9

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

$2 x^{2} + 9 + 9 = 0$

Step 10

Add $9$ and $9$ .

$2 x^{2} + 18 = 0$

Step 11

Factor $2$ out of $2 x^{2} + 18$ .

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Step 11.1

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

$2 (x^{2}) + 18 = 0$

Step 11.2

Factor $2$ out of $18$ .

$2 x^{2} + 2 \cdot 9 = 0$

Step 11.3

Factor $2$ out of $2 x^{2} + 2 \cdot 9$ .

$2 (x^{2} + 9) = 0$

Step 12

Divide each term in $2 (x^{2} + 9) = 0$ by $2$ and simplify.

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Step 12.1

Divide each term in $2 (x^{2} + 9) = 0$ by $2$ .

$\frac{2 (x^{2} + 9)}{2} = \frac{0}{2}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $2$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 9)}{2} = \frac{0}{2}$

Step 12.2.1.2

Divide $x^{2} + 9$ by $1$ .

$x^{2} + 9 = \frac{0}{2}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Divide $0$ by $2$ .

$x^{2} + 9 = 0$

Step 13

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 14

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 9}$

Step 15

Simplify $\pm \sqrt{- 9}$ .

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Step 15.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 15.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 15.3

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

$x = \pm i \cdot \sqrt{9}$

Step 15.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 15.5

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

$x = \pm i \cdot 3$

Step 15.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 16

The complete solution is the result of both the positive and negative portions of the solution.

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Step 16.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i$

Step 16.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

Step 16.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$