Finite Math Examples

$g (x) = 12 x - 36 \sqrt{x}$

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 $x$ .

$0 = 12 x - 36 \sqrt{x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $12 x - 36 \sqrt{x} = 0$ .

$12 x - 36 \sqrt{x} = 0$

Step 1.2.2

Subtract $12 x$ from both sides of the equation.

$- 36 \sqrt{x} = - 12 x$

Step 1.2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 36 \sqrt{x})}^{2} = {(- 12 x)}^{2}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(- 36 x^{\frac{1}{2}})}^{2} = {(- 12 x)}^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(- 36 x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.1

Apply the product rule to $- 36 x^{\frac{1}{2}}$ .

${(- 36)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 12 x)}^{2}$

Step 1.2.4.2.1.2

Raise $- 36$ to the power of $2$ .

$1296 {(x^{\frac{1}{2}})}^{2} = {(- 12 x)}^{2}$

Step 1.2.4.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1296 x^{\frac{1}{2} \cdot 2} = {(- 12 x)}^{2}$

Step 1.2.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.3.2.1

Cancel the common factor.

$1296 x^{\frac{1}{2} \cdot 2} = {(- 12 x)}^{2}$

Step 1.2.4.2.1.3.2.2

Rewrite the expression.

$1296 x^{1} = {(- 12 x)}^{2}$

Step 1.2.4.2.1.4

Simplify.

$1296 x = {(- 12 x)}^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Simplify ${(- 12 x)}^{2}$ .

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Step 1.2.4.3.1.1

Apply the product rule to $- 12 x$ .

$1296 x = {(- 12)}^{2} x^{2}$

Step 1.2.4.3.1.2

Raise $- 12$ to the power of $2$ .

$1296 x = 144 x^{2}$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Subtract $144 x^{2}$ from both sides of the equation.

$1296 x - 144 x^{2} = 0$

Step 1.2.5.2

Factor the left side of the equation.

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Step 1.2.5.2.1

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

$1296 u - 144 u^{2} = 0$

Step 1.2.5.2.2

Factor $144 u$ out of $1296 u - 144 u^{2}$ .

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Step 1.2.5.2.2.1

Factor $144 u$ out of $1296 u$ .

$144 u (9) - 144 u^{2} = 0$

Step 1.2.5.2.2.2

Factor $144 u$ out of $- 144 u^{2}$ .

$144 u (9) + 144 u (- u) = 0$

Step 1.2.5.2.2.3

Factor $144 u$ out of $144 u (9) + 144 u (- u)$ .

$144 u (9 - u) = 0$

Step 1.2.5.2.3

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

$144 x (9 - x) = 0$

Step 1.2.5.3

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

$x = 0$

$9 - x = 0$

Step 1.2.5.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5.5

Set $9 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.5.5.1

Set $9 - x$ equal to $0$ .

$9 - x = 0$

Step 1.2.5.5.2

Solve $9 - x = 0$ for $x$ .

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Step 1.2.5.5.2.1

Subtract $9$ from both sides of the equation.

$- x = - 9$

Step 1.2.5.5.2.2

Divide each term in $- x = - 9$ by $- 1$ and simplify.

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Step 1.2.5.5.2.2.1

Divide each term in $- x = - 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 9}{- 1}$

Step 1.2.5.5.2.2.2

Simplify the left side.

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Step 1.2.5.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 9}{- 1}$

Step 1.2.5.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 9}{- 1}$

Step 1.2.5.5.2.2.3

Simplify the right side.

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Step 1.2.5.5.2.2.3.1

Divide $- 9$ by $- 1$ .

$x = 9$

Step 1.2.5.6

The final solution is all the values that make $144 x (9 - x) = 0$ true.

$x = 0, 9$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 12 (0) - 36 \sqrt{0}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 12 (0) - 36 \sqrt{0}$

Step 2.2.2

Simplify $12 (0) - 36 \sqrt{0}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $12$ by $0$ .

$y = 0 - 36 \sqrt{0}$

Step 2.2.2.1.2

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

$y = 0 - 36 \sqrt{0^{2}}$

Step 2.2.2.1.3

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

$y = 0 - 36 \cdot 0$

Step 2.2.2.1.4

Multiply $- 36$ by $0$ .

$y = 0 + 0$

Step 2.2.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4