Precalculus Examples

$3 x^{2} - 4 x - 4$

Step 1

Write $3 x^{2} - 4 x - 4$ as an equation.

$y = 3 x^{2} - 4 x - 4$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = 3 x^{2} - 4 x - 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $3 x^{2} - 4 x - 4 = 0$ .

$3 x^{2} - 4 x - 4 = 0$

Step 2.2.2

Factor by grouping.

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Step 2.2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 4$ .

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Step 2.2.2.1.1

Factor $- 4$ out of $- 4 x$ .

$3 x^{2} - 4 x - 4 = 0$

Step 2.2.2.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$3 x^{2} + (2 - 6) x - 4 = 0$

Step 2.2.2.1.3

Apply the distributive property.

$3 x^{2} + 2 x - 6 x - 4 = 0$

Step 2.2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.2.1

Group the first two terms and the last two terms.

$(3 x^{2} + 2 x) - 6 x - 4 = 0$

Step 2.2.2.2.2

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

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

Step 2.2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

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

Step 2.2.3

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

$3 x + 2 = 0$

$x - 2 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

$3 x + 2 = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 2.2.4.2.2

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

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Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

Simplify the left side.

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Step 2.2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.4.2.2.3

Simplify the right side.

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Step 2.2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

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

$x - 2 = 0$

Step 2.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.6

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

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

Step 2.3

x-intercept(s) in point form.

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

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = 3 {(0)}^{2} - 4 \cdot 0 - 4$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 3 \cdot 0^{2} - 4 \cdot 0 - 4$

Step 3.2.2

Remove parentheses.

$y = 3 {(0)}^{2} - 4 \cdot 0 - 4$

Step 3.2.3

Simplify $3 {(0)}^{2} - 4 \cdot 0 - 4$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

$y = 3 \cdot 0 - 4 \cdot 0 - 4$

Step 3.2.3.1.2

Multiply $3$ by $0$ .

$y = 0 - 4 \cdot 0 - 4$

Step 3.2.3.1.3

Multiply $- 4$ by $0$ .

$y = 0 + 0 - 4$

Step 3.2.3.2

Simplify by adding and subtracting.

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Step 3.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 4$

Step 3.2.3.2.2

Subtract $4$ from $0$ .

$y = - 4$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 4)$

Step 4

List the intersections.

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

y-intercept(s): $(0, - 4)$

Step 5