Finite Math Examples

$x (75 - 5 x) - (122 + 16 x)$

Step 1

Write $x (75 - 5 x) - (122 + 16 x)$ as an equation.

$y = x (75 - 5 x) - (122 + 16 x)$

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 $x$ .

$0 = x (75 - 5 x) - (122 + 16 x)$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x (75 - 5 x) - (122 + 16 x) = 0$ .

$x (75 - 5 x) - (122 + 16 x) = 0$

Step 2.2.2

Simplify $x (75 - 5 x) - (122 + 16 x)$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Apply the distributive property.

$x \cdot 75 + x (- 5 x) - (122 + 16 x) = 0$

Step 2.2.2.1.2

Move $75$ to the left of $x$ .

$75 \cdot x + x (- 5 x) - (122 + 16 x) = 0$

Step 2.2.2.1.3

Rewrite using the commutative property of multiplication.

$75 \cdot x - 5 x \cdot x - (122 + 16 x) = 0$

Step 2.2.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.2.1.4.1

Move $x$ .

$75 x - 5 (x \cdot x) - (122 + 16 x) = 0$

Step 2.2.2.1.4.2

Multiply $x$ by $x$ .

$75 x - 5 x^{2} - (122 + 16 x) = 0$

Step 2.2.2.1.5

Apply the distributive property.

$75 x - 5 x^{2} - 1 \cdot 122 - (16 x) = 0$

Step 2.2.2.1.6

Multiply $- 1$ by $122$ .

$75 x - 5 x^{2} - 122 - (16 x) = 0$

Step 2.2.2.1.7

Multiply $16$ by $- 1$ .

$75 x - 5 x^{2} - 122 - 16 x = 0$

Step 2.2.2.2

Subtract $16 x$ from $75 x$ .

$59 x - 5 x^{2} - 122 = 0$

Step 2.2.3

Use the quadratic formula to find the solutions.

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

Step 2.2.4

Substitute the values $a = - 5$ , $b = 59$ , and $c = - 122$ into the quadratic formula and solve for $x$ .

$\frac{- 59 \pm \sqrt{59^{2} - 4 \cdot (- 5 \cdot - 122)}}{2 \cdot - 5}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Raise $59$ to the power of $2$ .

$x = \frac{- 59 \pm \sqrt{3481 - 4 \cdot - 5 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.5.1.2

Multiply $- 4 \cdot - 5 \cdot - 122$ .

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Step 2.2.5.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{3481 + 20 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.5.1.2.2

Multiply $20$ by $- 122$ .

$x = \frac{- 59 \pm \sqrt{3481 - 2440}}{2 \cdot - 5}$

Step 2.2.5.1.3

Subtract $2440$ from $3481$ .

$x = \frac{- 59 \pm \sqrt{1041}}{2 \cdot - 5}$

Step 2.2.5.2

Multiply $2$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{1041}}{- 10}$

Step 2.2.5.3

Simplify $\frac{- 59 \pm \sqrt{1041}}{- 10}$ .

$x = \frac{59 \pm \sqrt{1041}}{10}$

Step 2.2.6

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

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $59$ to the power of $2$ .

$x = \frac{- 59 \pm \sqrt{3481 - 4 \cdot - 5 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.6.1.2

Multiply $- 4 \cdot - 5 \cdot - 122$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{3481 + 20 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.6.1.2.2

Multiply $20$ by $- 122$ .

$x = \frac{- 59 \pm \sqrt{3481 - 2440}}{2 \cdot - 5}$

Step 2.2.6.1.3

Subtract $2440$ from $3481$ .

$x = \frac{- 59 \pm \sqrt{1041}}{2 \cdot - 5}$

Step 2.2.6.2

Multiply $2$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{1041}}{- 10}$

Step 2.2.6.3

Simplify $\frac{- 59 \pm \sqrt{1041}}{- 10}$ .

$x = \frac{59 \pm \sqrt{1041}}{10}$

Step 2.2.6.4

Change the $\pm$ to $+$ .

$x = \frac{59 + \sqrt{1041}}{10}$

Step 2.2.7

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

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise $59$ to the power of $2$ .

$x = \frac{- 59 \pm \sqrt{3481 - 4 \cdot - 5 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.7.1.2

Multiply $- 4 \cdot - 5 \cdot - 122$ .

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Step 2.2.7.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{3481 + 20 \cdot - 122}}{2 \cdot - 5}$

Step 2.2.7.1.2.2

Multiply $20$ by $- 122$ .

$x = \frac{- 59 \pm \sqrt{3481 - 2440}}{2 \cdot - 5}$

Step 2.2.7.1.3

Subtract $2440$ from $3481$ .

$x = \frac{- 59 \pm \sqrt{1041}}{2 \cdot - 5}$

Step 2.2.7.2

Multiply $2$ by $- 5$ .

$x = \frac{- 59 \pm \sqrt{1041}}{- 10}$

Step 2.2.7.3

Simplify $\frac{- 59 \pm \sqrt{1041}}{- 10}$ .

$x = \frac{59 \pm \sqrt{1041}}{10}$

Step 2.2.7.4

Change the $\pm$ to $-$ .

$x = \frac{59 - \sqrt{1041}}{10}$

Step 2.2.8

The final answer is the combination of both solutions.

$x = \frac{59 + \sqrt{1041}}{10}, \frac{59 - \sqrt{1041}}{10}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{59 + \sqrt{1041}}{10}, 0), (\frac{59 - \sqrt{1041}}{10}, 0)$

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 $y$ .

$y = (0) (75 - 5 \cdot 0) - (122 + 16 (0))$

Step 3.2

Solve the equation.

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Step 3.2.1

Multiply $0$ by $75 - 5 \cdot 0$ .

$y = 0 (75 - 5 \cdot 0) - (122 + 16 (0))$

Step 3.2.2

Remove parentheses.

$y = (0) (75 - 5 \cdot 0) - (122 + 16 (0))$

Step 3.2.3

Simplify $(0) (75 - 5 \cdot 0) - (122 + 16 (0))$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $- 5$ by $0$ .

$y = 0 (75 + 0) - (122 + 16 (0))$

Step 3.2.3.1.2

Add $75$ and $0$ .

$y = 0 \cdot 75 - (122 + 16 (0))$

Step 3.2.3.1.3

Multiply $0$ by $75$ .

$y = 0 - (122 + 16 (0))$

Step 3.2.3.1.4

Multiply $16$ by $0$ .

$y = 0 - (122 + 0)$

Step 3.2.3.1.5

Add $122$ and $0$ .

$y = 0 - 1 \cdot 122$

Step 3.2.3.1.6

Multiply $- 1$ by $122$ .

$y = 0 - 122$

Step 3.2.3.2

Subtract $122$ from $0$ .

$y = - 122$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 122)$

Step 4

List the intersections.

x-intercept(s): $(\frac{59 + \sqrt{1041}}{10}, 0), (\frac{59 - \sqrt{1041}}{10}, 0)$

y-intercept(s): $(0, - 122)$

Step 5