Algebra Examples

$(x + 3) (x - 3) = 16 - 2 x^{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $(x + 3) (x - 3)$ .

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Step 1.1.1.1

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

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Step 1.1.1.1.1

Apply the distributive property.

$x (x - 3) + 3 (x - 3) = 16 - 2 x^{2}$

Step 1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 3 (x - 3) = 16 - 2 x^{2}$

Step 1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3 = 16 - 2 x^{2}$

Step 1.1.1.2

Simplify terms.

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Step 1.1.1.2.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 1.1.1.2.1.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$x \cdot x - 3 x + 3 x + 3 \cdot - 3 = 16 - 2 x^{2}$

Step 1.1.1.2.1.2

Add $- 3 x$ and $3 x$ .

$x \cdot x + 0 + 3 \cdot - 3 = 16 - 2 x^{2}$

Step 1.1.1.2.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x + 3 \cdot - 3 = 16 - 2 x^{2}$

Step 1.1.1.2.2

Simplify each term.

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Step 1.1.1.2.2.1

Multiply $x$ by $x$ .

$x^{2} + 3 \cdot - 3 = 16 - 2 x^{2}$

Step 1.1.1.2.2.2

Multiply $3$ by $- 3$ .

$x^{2} - 9 = 16 - 2 x^{2}$

Step 1.2

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

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Step 1.2.1

Subtract $16$ from both sides of the equation.

$x^{2} - 9 - 16 = - 2 x^{2}$

Step 1.2.2

Add $2 x^{2}$ to both sides of the equation.

$x^{2} - 9 - 16 + 2 x^{2} = 0$

Step 1.3

Simplify $x^{2} - 9 - 16 + 2 x^{2}$ .

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Step 1.3.1

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

$3 x^{2} - 9 - 16 = 0$

Step 1.3.2

Subtract $16$ from $- 9$ .

$3 x^{2} - 25 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = 0$ , and $c = - 25$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (3 \cdot - 25)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 3 \cdot - 25}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot - 25$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{0 \pm \sqrt{0 - 12 \cdot - 25}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 25$ .

$x = \frac{0 \pm \sqrt{0 + 300}}{2 \cdot 3}$

Step 4.1.3

Add $0$ and $300$ .

$x = \frac{0 \pm \sqrt{300}}{2 \cdot 3}$

Step 4.1.4

Rewrite $300$ as $10^{2} \cdot 3$ .

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Step 4.1.4.1

Factor $100$ out of $300$ .

$x = \frac{0 \pm \sqrt{100 (3)}}{2 \cdot 3}$

Step 4.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{0 \pm \sqrt{10^{2} \cdot 3}}{2 \cdot 3}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{0 \pm 10 \sqrt{3}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{0 \pm 10 \sqrt{3}}{6}$

Step 4.3

Simplify $\frac{0 \pm 10 \sqrt{3}}{6}$ .

$x = \frac{\pm 5 \sqrt{3}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\pm 5 \sqrt{3}}{3}$

Decimal Form:

$x = 2.88675134 \dots, - 2.88675134 \dots$