Finite Math Examples

$f (x) = - 5 x^{2} + 30 x - 50$

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 $x$ .

$0 = - 5 x^{2} + 30 x - 50$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 5 x^{2} + 30 x - 50 = 0$ .

$- 5 x^{2} + 30 x - 50 = 0$

Step 1.2.2

Factor $- 5$ out of $- 5 x^{2} + 30 x - 50$ .

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Step 1.2.2.1

Factor $- 5$ out of $- 5 x^{2}$ .

$- 5 x^{2} + 30 x - 50 = 0$

Step 1.2.2.2

Factor $- 5$ out of $30 x$ .

$- 5 x^{2} - 5 (- 6 x) - 50 = 0$

Step 1.2.2.3

Factor $- 5$ out of $- 50$ .

$- 5 x^{2} - 5 (- 6 x) - 5 \cdot 10 = 0$

Step 1.2.2.4

Factor $- 5$ out of $- 5 (x^{2}) - 5 (- 6 x)$ .

$- 5 (x^{2} - 6 x) - 5 \cdot 10 = 0$

Step 1.2.2.5

Factor $- 5$ out of $- 5 (x^{2} - 6 x) - 5 (10)$ .

$- 5 (x^{2} - 6 x + 10) = 0$

Step 1.2.3

Divide each term in $- 5 (x^{2} - 6 x + 10) = 0$ by $- 5$ and simplify.

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Step 1.2.3.1

Divide each term in $- 5 (x^{2} - 6 x + 10) = 0$ by $- 5$ .

$\frac{- 5 (x^{2} - 6 x + 10)}{- 5} = \frac{0}{- 5}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 5 (x^{2} - 6 x + 10)}{- 5} = \frac{0}{- 5}$

Step 1.2.3.2.1.2

Divide $x^{2} - 6 x + 10$ by $1$ .

$x^{2} - 6 x + 10 = \frac{0}{- 5}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 5$ .

$x^{2} - 6 x + 10 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

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

Step 1.2.5

Substitute the values $a = 1$ , $b = - 6$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 1.2.6.1.3

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.6.1.7

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.6.1.8

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{6 \pm 2 i}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 i}{2}$

Step 1.2.6.3

Simplify $\frac{6 \pm 2 i}{2}$ .

$x = 3 \pm i$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 1.2.7.1.3

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.7.1.7

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.7.1.8

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{6 \pm 2 i}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 i}{2}$

Step 1.2.7.3

Simplify $\frac{6 \pm 2 i}{2}$ .

$x = 3 \pm i$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = 3 + i$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 1.2.8.1.3

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.8.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.8.1.7

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.8.1.8

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.8.1.9

Move $2$ to the left of $i$ .

$x = \frac{6 \pm 2 i}{2 \cdot 1}$

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 i}{2}$

Step 1.2.8.3

Simplify $\frac{6 \pm 2 i}{2}$ .

$x = 3 \pm i$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = 3 - i$

Step 1.2.9

The final answer is the combination of both solutions.

$x = 3 + i, 3 - i$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

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 $y$ .

$y = - 5 {(0)}^{2} + 30 (0) - 50$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 5 \cdot 0^{2} + 30 (0) - 50$

Step 2.2.2

Remove parentheses.

$y = - 5 {(0)}^{2} + 30 (0) - 50$

Step 2.2.3

Simplify $- 5 {(0)}^{2} + 30 (0) - 50$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - 5 \cdot 0 + 30 (0) - 50$

Step 2.2.3.1.2

Multiply $- 5$ by $0$ .

$y = 0 + 30 (0) - 50$

Step 2.2.3.1.3

Multiply $30$ by $0$ .

$y = 0 + 0 - 50$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 50$

Step 2.2.3.2.2

Subtract $50$ from $0$ .

$y = - 50$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 50)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 50)$

Step 4