Finite Math Examples

$y = - \frac{1}{3} \cdot {(x + 3)}^{2} + 6$

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 $x$ .

$0 = - \frac{1}{3} \cdot {(x + 3)}^{2} + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{3} \cdot {(x + 3)}^{2} + 6 = 0$ .

$- \frac{1}{3} \cdot {(x + 3)}^{2} + 6 = 0$

Step 1.2.2

Subtract $6$ from both sides of the equation.

$- \frac{1}{3} \cdot {(x + 3)}^{2} = - 6$

Step 1.2.3

Combine ${(x + 3)}^{2}$ and $\frac{1}{3}$ .

$- \frac{{(x + 3)}^{2}}{3} = - 6$

Step 1.2.4

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{{(x + 3)}^{2}}{3}) = - 3 \cdot - 6$

Step 1.2.5

Simplify both sides of the equation.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Simplify $- 3 (- \frac{{(x + 3)}^{2}}{3})$ .

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Step 1.2.5.1.1.1

Cancel the common factor of $3$ .

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Step 1.2.5.1.1.1.1

Move the leading negative in $- \frac{{(x + 3)}^{2}}{3}$ into the numerator.

$- 3 \frac{- {(x + 3)}^{2}}{3} = - 3 \cdot - 6$

Step 1.2.5.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- {(x + 3)}^{2}}{3} = - 3 \cdot - 6$

Step 1.2.5.1.1.1.3

Cancel the common factor.

$3 \cdot - 1 \frac{- {(x + 3)}^{2}}{3} = - 3 \cdot - 6$

Step 1.2.5.1.1.1.4

Rewrite the expression.

$- - {(x + 3)}^{2} = - 3 \cdot - 6$

Step 1.2.5.1.1.2

Multiply.

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Step 1.2.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 {(x + 3)}^{2} = - 3 \cdot - 6$

Step 1.2.5.1.1.2.2

Multiply ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = - 3 \cdot - 6$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $- 3$ by $- 6$ .

${(x + 3)}^{2} = 18$

Step 1.2.6

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

$x + 3 = \pm \sqrt{18}$

Step 1.2.7

Simplify $\pm \sqrt{18}$ .

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Step 1.2.7.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.2.7.1.1

Factor $9$ out of $18$ .

$x + 3 = \pm \sqrt{9 (2)}$

Step 1.2.7.1.2

Rewrite $9$ as $3^{2}$ .

$x + 3 = \pm \sqrt{3^{2} \cdot 2}$

Step 1.2.7.2

Pull terms out from under the radical.

$x + 3 = \pm 3 \sqrt{2}$

Step 1.2.8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.8.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 3 = 3 \sqrt{2}$

Step 1.2.8.2

Subtract $3$ from both sides of the equation.

$x = 3 \sqrt{2} - 3$

Step 1.2.8.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 3 = - 3 \sqrt{2}$

Step 1.2.8.4

Subtract $3$ from both sides of the equation.

$x = - 3 \sqrt{2} - 3$

Step 1.2.8.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 \sqrt{2} - 3, - 3 \sqrt{2} - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3 \sqrt{2} - 3, 0), (- 3 \sqrt{2} - 3, 0)$

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 $y$ .

$y = - \frac{1}{3} \cdot {((0) + 3)}^{2} + 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{1}{3} \cdot {(0 + 3)}^{2} + 6$

Step 2.2.2

Remove parentheses.

$y = - \frac{1}{3} \cdot {((0) + 3)}^{2} + 6$

Step 2.2.3

Simplify $- \frac{1}{3} \cdot {((0) + 3)}^{2} + 6$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $3$ .

$y = - \frac{1}{3} \cdot 3^{2} + 6$

Step 2.2.3.1.2

Cancel the common factor of $3$ .

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Step 2.2.3.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y = \frac{- 1}{3} \cdot 3^{2} + 6$

Step 2.2.3.1.2.2

Factor $3$ out of $3^{2}$ .

$y = \frac{- 1}{3} \cdot (3 \cdot 3) + 6$

Step 2.2.3.1.2.3

Cancel the common factor.

$y = \frac{- 1}{3} \cdot (3 \cdot 3) + 6$

Step 2.2.3.1.2.4

Rewrite the expression.

$y = - 1 \cdot 3 + 6$

Step 2.2.3.1.3

Multiply $- 1$ by $3$ .

$y = - 3 + 6$

Step 2.2.3.2

Add $- 3$ and $6$ .

$y = 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 3

List the intersections.

x-intercept(s): $(3 \sqrt{2} - 3, 0), (- 3 \sqrt{2} - 3, 0)$

y-intercept(s): $(0, 3)$

Step 4