Calculus Examples

$y = 3 x^{2} + 12 x + 12$

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 $x$ .

$0 = 3 x^{2} + 12 x + 12$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 x^{2} + 12 x + 12 = 0$ .

$3 x^{2} + 12 x + 12 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $3$ out of $3 x^{2} + 12 x + 12$ .

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Step 1.2.2.1.1

Factor $3$ out of $3 x^{2}$ .

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

Step 1.2.2.1.2

Factor $3$ out of $12 x$ .

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

Step 1.2.2.1.3

Factor $3$ out of $12$ .

$3 x^{2} + 3 (4 x) + 3 \cdot 4 = 0$

Step 1.2.2.1.4

Factor $3$ out of $3 x^{2} + 3 (4 x)$ .

$3 (x^{2} + 4 x) + 3 \cdot 4 = 0$

Step 1.2.2.1.5

Factor $3$ out of $3 (x^{2} + 4 x) + 3 \cdot 4$ .

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

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

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

$4 x = 2 \cdot x \cdot 2$

Step 1.2.2.2.3

Rewrite the polynomial.

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

Step 1.2.2.2.4

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

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

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $3$ .

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

Step 1.2.4

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

$x + 2 = 0$

Step 1.2.5

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \cdot 0^{2} + 12 (0) + 12$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $3 {(0)}^{2} + 12 (0) + 12$ .

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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 $0$ .

$y = 3 \cdot 0 + 12 (0) + 12$

Step 2.2.3.1.2

Multiply $3$ by $0$ .

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

Step 2.2.3.1.3

Multiply $12$ by $0$ .

$y = 0 + 0 + 12$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 12$

Step 2.2.3.2.2

Add $0$ and $12$ .

$y = 12$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 12)$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0)$

y-intercept(s): $(0, 12)$

Step 4