Algebra Examples

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

Step 1

Set $- 4 x^{2} - 9 + 12 x$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

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

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Step 2.1.1.1

Move $- 9$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

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

Step 2.1.1.5

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

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

Step 2.1.1.6

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

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

Step 2.1.2

Factor using the perfect square rule.

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Step 2.1.2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Rewrite the polynomial.

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

Step 2.1.2.5

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

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

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $- 1$ .

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

Step 2.3

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

$2 x - 3 = 0$

Step 2.4

Solve for $x$ .

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Step 2.4.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 3