Calculus Examples

$\frac{9 x^{2} (4 x - 3)}{{(4 x - 3)}^{2} (x + 7) (x - 9)}$

Step 1

Write $\frac{9 x^{2} (4 x - 3)}{{(4 x - 3)}^{2} (x + 7) (x - 9)}$ as an equation.

$y = \frac{9 x^{2} (4 x - 3)}{{(4 x - 3)}^{2} (x + 7) (x - 9)}$

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 $x$ .

$0 = \frac{9 x^{2} (4 x - 3)}{{(4 x - 3)}^{2} (x + 7) (x - 9)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$9 x^{2} (4 x - 3) = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

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

$x^{2} = 0$

$4 x - 3 = 0$

Step 2.2.2.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.2.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.2.2.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.2.2.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.2.2.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.2.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.2.3

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 2.2.2.3.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 2.2.2.3.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 2.2.2.3.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 2.2.2.3.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.2.2.3.2.2.2

Simplify the left side.

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Step 2.2.2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.2.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 2.2.2.4

The final solution is all the values that make $9 x^{2} (4 x - 3) = 0$ true.

$x = 0, \frac{3}{4}$

Step 2.2.3

Exclude the solutions that do not make $0 = \frac{9 x^{2} (4 x - 3)}{{(4 x - 3)}^{2} (x + 7) (x - 9)}$ true.

$x = 0$

Step 2.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{9 {(0)}^{2} (4 (0) - 3)}{{(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{9 \cdot (0^{2} (4 (0) - 3))}{{(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)}$

Step 3.2.2

Remove parentheses.

$y = \frac{9 \cdot (0^{2} (4 (0) - 3))}{{(4 (0) - 3)}^{2} (0 + 7) ((0) - 9)}$

Step 3.2.3

Remove parentheses.

$y = \frac{9 \cdot (0^{2} (4 (0) - 3))}{{(4 (0) - 3)}^{2} (0 + 7) (0 - 9)}$

Step 3.2.4

Remove parentheses.

$y = \frac{9 {(0)}^{2} (4 (0) - 3)}{{(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)}$

Step 3.2.5

Simplify $\frac{9 {(0)}^{2} (4 (0) - 3)}{{(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)}$ .

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Step 3.2.5.1

Reduce the expression by cancelling the common factors.

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Step 3.2.5.1.1

Cancel the common factor of $9$ and $(0) - 9$ .

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Step 3.2.5.1.1.1

Factor $9$ out of $9 {(0)}^{2} (4 (0) - 3)$ .

$y = \frac{9 ({(0)}^{2} (4 (0) - 3))}{{(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)}$

Step 3.2.5.1.1.2

Cancel the common factors.

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Step 3.2.5.1.1.2.1

Factor $9$ out of ${(4 (0) - 3)}^{2} ((0) + 7) ((0) - 9)$ .

$y = \frac{9 ({(0)}^{2} (4 (0) - 3))}{9 ({(4 (0) - 3)}^{2} ((0) + 7) (0 - 1))}$

Step 3.2.5.1.1.2.2

Cancel the common factor.

$y = \frac{9 ({(0)}^{2} (4 (0) - 3))}{9 ({(4 (0) - 3)}^{2} ((0) + 7) (0 - 1))}$

Step 3.2.5.1.1.2.3

Rewrite the expression.

$y = \frac{{(0)}^{2} (4 (0) - 3)}{{(4 (0) - 3)}^{2} ((0) + 7) (0 - 1)}$

Step 3.2.5.1.2

Cancel the common factor of $4 (0) - 3$ and ${(4 (0) - 3)}^{2}$ .

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Step 3.2.5.1.2.1

Factor $4 (0) - 3$ out of ${(0)}^{2} (4 (0) - 3)$ .

$y = \frac{(4 (0) - 3) {(0)}^{2}}{{(4 (0) - 3)}^{2} ((0) + 7) (0 - 1)}$

Step 3.2.5.1.2.2

Cancel the common factors.

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Step 3.2.5.1.2.2.1

Factor $4 (0) - 3$ out of ${(4 (0) - 3)}^{2} ((0) + 7) (0 - 1)$ .

$y = \frac{(4 (0) - 3) {(0)}^{2}}{(4 \cdot 0 - 3) (((4 \cdot 0 - 3) ((0) + 7)) (0 - 1))}$

Step 3.2.5.1.2.2.2

Cancel the common factor.

$y = \frac{(4 (0) - 3) {(0)}^{2}}{(4 \cdot 0 - 3) (((4 \cdot 0 - 3) ((0) + 7)) (0 - 1))}$

Step 3.2.5.1.2.2.3

Rewrite the expression.

$y = \frac{{(0)}^{2}}{((4 \cdot 0 - 3) ((0) + 7)) (0 - 1)}$

Step 3.2.5.1.3

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

$y = \frac{0}{(4 \cdot 0 - 3) (0 + 7) (0 - 1)}$

Step 3.2.5.2

Simplify the denominator.

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Step 3.2.5.2.1

Multiply $4$ by $0$ .

$y = \frac{0}{(0 - 3) (0 + 7) (0 - 1)}$

Step 3.2.5.2.2

Subtract $3$ from $0$ .

$y = \frac{0}{- 3 (0 + 7) (0 - 1)}$

Step 3.2.5.2.3

Add $0$ and $7$ .

$y = \frac{0}{- 3 \cdot 7 (0 - 1)}$

Step 3.2.5.2.4

Multiply $- 3$ by $7$ .

$y = \frac{0}{- 21 (0 - 1)}$

Step 3.2.5.2.5

Subtract $1$ from $0$ .

$y = \frac{0}{- 21 \cdot - 1}$

Step 3.2.5.3

Simplify the expression.

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Step 3.2.5.3.1

Multiply $- 21$ by $- 1$ .

$y = \frac{0}{21}$

Step 3.2.5.3.2

Divide $0$ by $21$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

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

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

Step 5