Finite Math Examples

$y - 2 = \frac{4}{3} \cdot (x (x - 4))$

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 $x$ .

$(0) - 2 = \frac{4}{3} \cdot (x (x - 4))$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{4}{3} \cdot (x (x - 4)) = (0) - 2$ .

$\frac{4}{3} \cdot (x (x - 4)) = (0) - 2$

Step 1.2.2

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} (\frac{4}{3} \cdot (x (x - 4))) = \frac{3}{4} ((0) - 2)$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $\frac{3}{4} (\frac{4}{3} \cdot (x (x - 4)))$ .

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Step 1.2.3.1.1.1

Apply the distributive property.

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

Step 1.2.3.1.1.2

Simplify the expression.

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Step 1.2.3.1.1.2.1

Multiply $x$ by $x$ .

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

Step 1.2.3.1.1.2.2

Move $- 4$ to the left of $x$ .

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

Step 1.2.3.1.1.3

Apply the distributive property.

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

Step 1.2.3.1.1.4

Combine $\frac{4}{3}$ and $x^{2}$ .

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

Step 1.2.3.1.1.5

Multiply $\frac{4}{3} (- 4 x)$ .

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Step 1.2.3.1.1.5.1

Combine $- 4$ and $\frac{4}{3}$ .

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

Step 1.2.3.1.1.5.2

Multiply $- 4$ by $4$ .

$\frac{3}{4} (\frac{4 x^{2}}{3} + \frac{- 16}{3} x) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.5.3

Combine $\frac{- 16}{3}$ and $x$ .

$\frac{3}{4} (\frac{4 x^{2}}{3} + \frac{- 16 x}{3}) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6

Simplify terms.

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Step 1.2.3.1.1.6.1

Move the negative in front of the fraction.

$\frac{3}{4} (\frac{4 x^{2}}{3} - \frac{16 x}{3}) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.2

Apply the distributive property.

$\frac{3}{4} \cdot \frac{4 x^{2}}{3} + \frac{3}{4} (- \frac{16 x}{3}) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.3

Combine.

$\frac{3 (4 x^{2})}{4 \cdot 3} + \frac{3}{4} (- \frac{16 x}{3}) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.4

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.6.4.1

Move the leading negative in $- \frac{16 x}{3}$ into the numerator.

$\frac{3 (4 x^{2})}{4 \cdot 3} + \frac{3}{4} \cdot \frac{- 16 x}{3} = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.4.2

Cancel the common factor.

$\frac{3 (4 x^{2})}{4 \cdot 3} + \frac{3}{4} \cdot \frac{- 16 x}{3} = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.4.3

Rewrite the expression.

$\frac{3 (4 x^{2})}{4 \cdot 3} + \frac{1}{4} (- 16 x) = \frac{3}{4} ((0) - 2)$

Step 1.2.3.1.1.6.5

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.6.5.1

Factor $4$ out of $- 16 x$ .

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

Step 1.2.3.1.1.6.5.2

Cancel the common factor.

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

Step 1.2.3.1.1.6.5.3

Rewrite the expression.

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

Step 1.2.3.1.1.7

Simplify each term.

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Step 1.2.3.1.1.7.1

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.7.1.1

Cancel the common factor.

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

Step 1.2.3.1.1.7.1.2

Rewrite the expression.

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

Step 1.2.3.1.1.7.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.7.2.1

Cancel the common factor.

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

Step 1.2.3.1.1.7.2.2

Divide $x^{2}$ by $1$ .

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Subtract $2$ from $0$ .

$x^{2} - 4 x = \frac{3}{4} \cdot - 2$

Step 1.2.3.2.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.2.1

Factor $2$ out of $4$ .

$x^{2} - 4 x = \frac{3}{2 (2)} \cdot - 2$

Step 1.2.3.2.1.2.2

Factor $2$ out of $- 2$ .

$x^{2} - 4 x = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 1.2.3.2.1.2.3

Cancel the common factor.

$x^{2} - 4 x = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 1.2.3.2.1.2.4

Rewrite the expression.

$x^{2} - 4 x = \frac{3}{2} \cdot - 1$

Step 1.2.3.2.1.3

Combine $\frac{3}{2}$ and $- 1$ .

$x^{2} - 4 x = \frac{3 \cdot - 1}{2}$

Step 1.2.3.2.1.4

Simplify the expression.

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Step 1.2.3.2.1.4.1

Multiply $3$ by $- 1$ .

$x^{2} - 4 x = \frac{- 3}{2}$

Step 1.2.3.2.1.4.2

Move the negative in front of the fraction.

$x^{2} - 4 x = - \frac{3}{2}$

Step 1.2.4

Add $\frac{3}{2}$ to both sides of the equation.

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

Step 1.2.5

Multiply through by the least common denominator $2$ , then simplify.

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Step 1.2.5.1

Apply the distributive property.

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

Step 1.2.5.2

Simplify.

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Step 1.2.5.2.1

Multiply $- 4$ by $2$ .

$2 x^{2} - 8 x + 2 (\frac{3}{2}) = 0$

Step 1.2.5.2.2

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.1

Cancel the common factor.

$2 x^{2} - 8 x + 2 (\frac{3}{2}) = 0$

Step 1.2.5.2.2.2

Rewrite the expression.

$2 x^{2} - 8 x + 3 = 0$

Step 1.2.6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.7

Substitute the values $a = 2$ , $b = - 8$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 1.2.8

Simplify.

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{8 \pm \sqrt{64 - 8 \cdot 3}}{2 \cdot 2}$

Step 1.2.8.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 2}$

Step 1.2.8.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 2}$

Step 1.2.8.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 1.2.8.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 2}$

Step 1.2.8.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 2}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 2}$

Step 1.2.8.2

Multiply $2$ by $2$ .

$x = \frac{8 \pm 2 \sqrt{10}}{4}$

Step 1.2.8.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{4}$ .

$x = \frac{4 \pm \sqrt{10}}{2}$

Step 1.2.9

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.9.1

Simplify the numerator.

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Step 1.2.9.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 1.2.9.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.2.9.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{8 \pm \sqrt{64 - 8 \cdot 3}}{2 \cdot 2}$

Step 1.2.9.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 2}$

Step 1.2.9.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 2}$

Step 1.2.9.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 1.2.9.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 2}$

Step 1.2.9.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 2}$

Step 1.2.9.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 2}$

Step 1.2.9.2

Multiply $2$ by $2$ .

$x = \frac{8 \pm 2 \sqrt{10}}{4}$

Step 1.2.9.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{4}$ .

$x = \frac{4 \pm \sqrt{10}}{2}$

Step 1.2.9.4

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{10}}{2}$

Step 1.2.10

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.10.1

Simplify the numerator.

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Step 1.2.10.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 1.2.10.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.2.10.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{8 \pm \sqrt{64 - 8 \cdot 3}}{2 \cdot 2}$

Step 1.2.10.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 2}$

Step 1.2.10.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 2}$

Step 1.2.10.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 1.2.10.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 2}$

Step 1.2.10.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 2}$

Step 1.2.10.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 2}$

Step 1.2.10.2

Multiply $2$ by $2$ .

$x = \frac{8 \pm 2 \sqrt{10}}{4}$

Step 1.2.10.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{4}$ .

$x = \frac{4 \pm \sqrt{10}}{2}$

Step 1.2.10.4

Change the $\pm$ to $-$ .

$x = \frac{4 - \sqrt{10}}{2}$

Step 1.2.11

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{10}}{2}, \frac{4 - \sqrt{10}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{4 + \sqrt{10}}{2}, 0), (\frac{4 - \sqrt{10}}{2}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Factor $3$ out of $(0) ((0) - 4)$ .

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

Step 2.2.1.1.2

Cancel the common factor.

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

Step 2.2.1.1.3

Rewrite the expression.

$y - 2 = 4 \cdot ((0) ((0) - 4))$

Step 2.2.1.2

Multiply by zero.

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Step 2.2.1.2.1

Multiply $0$ by $(0) - 4$ .

$y - 2 = 4 \cdot 0$

Step 2.2.1.2.2

Multiply $4$ by $0$ .

$y - 2 = 0$

Step 2.2.2

Add $2$ to both sides of the equation.

$y = 2$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(\frac{4 + \sqrt{10}}{2}, 0), (\frac{4 - \sqrt{10}}{2}, 0)$

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

Step 4