Precalculus Examples

$8 x^{2} + {(x + 3)}^{2} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$8 x^{2} + {(x + 3)}^{2} - 9 = 0$

Step 2

Simplify $8 x^{2} + {(x + 3)}^{2} - 9$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$8 x^{2} + (x + 3) (x + 3) - 9 = 0$

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

$8 x^{2} + x (x + 3) + 3 (x + 3) - 9 = 0$

Step 2.1.2.2

Apply the distributive property.

$8 x^{2} + x \cdot x + x \cdot 3 + 3 (x + 3) - 9 = 0$

Step 2.1.2.3

Apply the distributive property.

$8 x^{2} + x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 - 9 = 0$

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 $x$ .

$8 x^{2} + x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 - 9 = 0$

Step 2.1.3.1.2

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

$8 x^{2} + x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 - 9 = 0$

Step 2.1.3.1.3

Multiply $3$ by $3$ .

$8 x^{2} + x^{2} + 3 x + 3 x + 9 - 9 = 0$

Step 2.1.3.2

Add $3 x$ and $3 x$ .

$8 x^{2} + x^{2} + 6 x + 9 - 9 = 0$

Step 2.2

Combine the opposite terms in $8 x^{2} + x^{2} + 6 x + 9 - 9$ .

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Step 2.2.1

Subtract $9$ from $9$ .

$8 x^{2} + x^{2} + 6 x + 0 = 0$

Step 2.2.2

Add $8 x^{2} + x^{2} + 6 x$ and $0$ .

$8 x^{2} + x^{2} + 6 x = 0$

Step 2.3

Add $8 x^{2}$ and $x^{2}$ .

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

Step 3

Factor $3 x$ out of $9 x^{2} + 6 x$ .

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Step 3.1

Factor $3 x$ out of $9 x^{2}$ .

$3 x (3 x) + 6 x = 0$

Step 3.2

Factor $3 x$ out of $6 x$ .

$3 x (3 x) + 3 x (2) = 0$

Step 3.3

Factor $3 x$ out of $3 x (3 x) + 3 x (2)$ .

$3 x (3 x + 2) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 x + 2 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $3 x + 2$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Step 6.2

Solve $3 x + 2 = 0$ for $x$ .

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Step 6.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 6.2.2

Divide each term in $3 x = - 2$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 7

The final solution is all the values that make $3 x (3 x + 2) = 0$ true.

$x = 0, - \frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{2}{3}$

Decimal Form:

$x = 0, - 0.\overline{6}$