Algebra Examples

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

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 $x$ .

$0 = 3 - \frac{1}{2} x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 - \frac{1}{2} x^{2} = 0$ .

$3 - \frac{1}{2} x^{2} = 0$

Step 1.2.2

Combine $x^{2}$ and $\frac{1}{2}$ .

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

Step 1.2.3

Subtract $3$ from both sides of the equation.

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

Step 1.2.4

Multiply both sides of the equation by $- 2$ .

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

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

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Step 1.2.5.1.1.1

Cancel the common factor of $2$ .

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Step 1.2.5.1.1.1.1

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

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

Step 1.2.5.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 1.2.5.1.1.1.3

Cancel the common factor.

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

Step 1.2.5.1.1.1.4

Rewrite the expression.

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

Step 1.2.5.1.1.2

Multiply.

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Step 1.2.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.1.1.2.2

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

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

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $- 2$ by $- 3$ .

$x^{2} = 6$

Step 1.2.6

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

$x = \pm \sqrt{6}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x = \sqrt{6}$

Step 1.2.7.2

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

$x = - \sqrt{6}$

Step 1.2.7.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{6}, 0), (- \sqrt{6}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 - \frac{1}{2} \cdot 0^{2}$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

$y = 3 - \frac{1}{2} \cdot 0$

Step 2.2.3.1.2

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 2.2.3.1.2.1

Multiply $0$ by $- 1$ .

$y = 3 + 0 (\frac{1}{2})$

Step 2.2.3.1.2.2

Multiply $0$ by $\frac{1}{2}$ .

$y = 3 + 0$

Step 2.2.3.2

Add $3$ and $0$ .

$y = 3$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(\sqrt{6}, 0), (- \sqrt{6}, 0)$

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

Step 4