Algebra Examples

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

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

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 9 \cdot 2 = 18

$a \cdot c = 9 \cdot 2 = 18$ and whose sum is

b = 9

$b = 9$ .

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Step 1.1.1

Factor

9

$9$ out of

9 x

$9 x$ .

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

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

Step 1.1.2

Rewrite

9

$9$ as

3

$3$ plus

6

$6$

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

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

Step 1.1.3

Apply the distributive property.

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

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

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

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

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

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

3 x (3 x + 1) + 2 (3 x + 1) = 0

$3 x (3 x + 1) + 2 (3 x + 1) = 0$

3 x (3 x + 1) + 2 (3 x + 1) = 0

$3 x (3 x + 1) + 2 (3 x + 1) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

3 x + 1

$3 x + 1$ .

(3 x + 1) (3 x + 2) = 0

$(3 x + 1) (3 x + 2) = 0$

(3 x + 1) (3 x + 2) = 0

$(3 x + 1) (3 x + 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

3 x + 1 = 0

$3 x + 1 = 0$

3 x + 2 = 0

$3 x + 2 = 0$

Step 3

Set

3 x + 1

$3 x + 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.1

Set

3 x + 1

$3 x + 1$ equal to

0

$0$ .

3 x + 1 = 0

$3 x + 1 = 0$

Step 3.2

Solve

3 x + 1 = 0

$3 x + 1 = 0$ for

x

$x$ .

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Step 3.2.1

Subtract

1

$1$ from both sides of the equation.

3 x = - 1

$3 x = - 1$

Step 3.2.2

Divide each term in

3 x = - 1

$3 x = - 1$ by

3

$3$ and simplify.

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Step 3.2.2.1

Divide each term in

3 x = - 1

$3 x = - 1$ by

3

$3$ .

\frac{3 x}{3} = \frac{- 1}{3}

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of

3

$3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

\frac{3 x}{3} = \frac{- 1}{3}

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 1}{3}

$x = \frac{- 1}{3}$

x = \frac{- 1}{3}

$x = \frac{- 1}{3}$

x = \frac{- 1}{3}

$x = \frac{- 1}{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

x = - \frac{1}{3}

$x = - \frac{1}{3}$

x = - \frac{1}{3}

$x = - \frac{1}{3}$

x = - \frac{1}{3}

$x = - \frac{1}{3}$

x = - \frac{1}{3}

$x = - \frac{1}{3}$

x = - \frac{1}{3}

$x = - \frac{1}{3}$

Step 4

Set

3 x + 2

$3 x + 2$ equal to

0

$0$ and solve for

x

$x$ .

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Step 4.1

Set

3 x + 2

$3 x + 2$ equal to

0

$0$ .

3 x + 2 = 0

$3 x + 2 = 0$

Step 4.2

Solve

3 x + 2 = 0

$3 x + 2 = 0$ for

x

$x$ .

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Step 4.2.1

Subtract

2

$2$ from both sides of the equation.

3 x = - 2

$3 x = - 2$

Step 4.2.2

Divide each term in

3 x = - 2

$3 x = - 2$ by

3

$3$ and simplify.

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Step 4.2.2.1

Divide each term in

3 x = - 2

$3 x = - 2$ by

3

$3$ .

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

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

3

$3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

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

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

Step 4.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 2}{3}

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

x = \frac{- 2}{3}

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

x = \frac{- 2}{3}

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

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

Step 5

The final solution is all the values that make

(3 x + 1) (3 x + 2) = 0

$(3 x + 1) (3 x + 2) = 0$ true.

x = - \frac{1}{3}, - \frac{2}{3}

$x = - \frac{1}{3}, - \frac{2}{3}$