Algebra Examples

$- 9 x^{2} + 12 x - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $- 1$ out of $- 9 x^{2} + 12 x - 4$ .

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Step 1.1.1

Factor $- 1$ out of $- 9 x^{2}$ .

$- (9 x^{2}) + 12 x - 4 = 0$

Step 1.1.2

Factor $- 1$ out of $12 x$ .

$- (9 x^{2}) - (- 12 x) - 4 = 0$

Step 1.1.3

Rewrite $- 4$ as $- 1 (4)$ .

$- (9 x^{2}) - (- 12 x) - 1 \cdot 4 = 0$

Step 1.1.4

Factor $- 1$ out of $- (9 x^{2}) - (- 12 x)$ .

$- (9 x^{2} - 12 x) - 1 \cdot 4 = 0$

Step 1.1.5

Factor $- 1$ out of $- (9 x^{2} - 12 x) - 1 (4)$ .

$- (9 x^{2} - 12 x + 4) = 0$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$- ({(3 x)}^{2} - 12 x + 4) = 0$

Step 1.2.2

Rewrite $4$ as $2^{2}$ .

$- ({(3 x)}^{2} - 12 x + 2^{2}) = 0$

Step 1.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot (3 x) \cdot 2$

Step 1.2.4

Rewrite the polynomial.

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

Step 1.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 x$ and $b = 2$ .

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

Step 2

Divide each term in $- {(3 x - 2)}^{2} = 0$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- {(3 x - 2)}^{2} = 0$ by $- 1$ .

$\frac{- {(3 x - 2)}^{2}}{- 1} = \frac{0}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{{(3 x - 2)}^{2}}{1} = \frac{0}{- 1}$

Step 2.2.2

Divide ${(3 x - 2)}^{2}$ by $1$ .

${(3 x - 2)}^{2} = \frac{0}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 1$ .

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

Step 3

Set the $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 5