Calculus Examples

$y = 12 x - 4 x^{2} - 9$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$12 u - 4 u^{2} - 9 = 0$

Step 1.2.2.2

Factor by grouping.

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Step 1.2.2.2.1

Reorder terms.

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

Step 1.2.2.2.2

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 $a \cdot c = - 4 \cdot - 9 = 36$ and whose sum is $b = 12$ .

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Step 1.2.2.2.2.1

Factor $12$ out of $12 u$ .

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

Step 1.2.2.2.2.2

Rewrite $12$ as $6$ plus $6$

$- 4 u^{2} + (6 + 6) u - 9 = 0$

Step 1.2.2.2.2.3

Apply the distributive property.

$- 4 u^{2} + 6 u + 6 u - 9 = 0$

Step 1.2.2.2.3

Factor out the greatest common factor from each group.

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Step 1.2.2.2.3.1

Group the first two terms and the last two terms.

$(- 4 u^{2} + 6 u) + 6 u - 9 = 0$

Step 1.2.2.2.3.2

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

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

Step 1.2.2.2.4

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

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

Step 1.2.2.3

Replace all occurrences of $u$ with $x$ .

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

Step 1.2.3

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

$- 2 x + 3 = 0$

$2 x - 3 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$- 2 x + 3 = 0$

Step 1.2.4.2

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

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Step 1.2.4.2.1

Subtract $3$ from both sides of the equation.

$- 2 x = - 3$

Step 1.2.4.2.2

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

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Step 1.2.4.2.2.1

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

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

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 1.2.5

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

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $12$ by $0$ .

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

Step 2.2.3.1.2

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

$y = 0 - 4 \cdot 0 - 9$

Step 2.2.3.1.3

Multiply $- 4$ 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