Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4 x^{2} + 12 x - 9 = 0$ .

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

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

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

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

Step 1.2.2.1.3

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

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

Step 1.2.2.1.4

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

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

Step 1.2.2.1.5

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

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

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

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

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

Step 1.2.2.2.3

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 1.2.2.2.4

Rewrite the polynomial.

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

Step 1.2.2.2.5

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 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2

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

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 1$ .

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

Step 1.2.4

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

$2 x - 3 = 0$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

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

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 4 \cdot 0^{2} + 12 (0) - 9$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $- 4 {(0)}^{2} + 12 (0) - 9$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - 4 \cdot 0 + 12 (0) - 9$

Step 2.2.3.1.2

Multiply $- 4$ by $0$ .

$y = 0 + 12 (0) - 9$

Step 2.2.3.1.3

Multiply $12$ by $0$ .

$y = 0 + 0 - 9$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 9$

Step 2.2.3.2.2

Subtract $9$ from $0$ .

$y = - 9$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 9)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3}{2}, 0)$

y-intercept(s): $(0, - 9)$

Step 4