Find the x and y Intercepts f(x)=-1/2(x-5)^2-3

$f (x) = - \frac{1}{2} {(x - 5)}^{2} - 3$

Find the x-intercepts.

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

$0 = - \frac{1}{2} \cdot {(x - 5)}^{2} - 3$

Solve the equation.

Rewrite the equation as $- \frac{1}{2} \cdot {(x - 5)}^{2} - 3 = 0$ .

$- \frac{1}{2} \cdot {(x - 5)}^{2} - 3 = 0$

Add $3$ to both sides of the equation.

$- \frac{1}{2} \cdot {(x - 5)}^{2} = 3$

Combine ${(x - 5)}^{2}$ and $\frac{1}{2}$ .

$- \frac{{(x - 5)}^{2}}{2} = 3$

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

$- 2 (- \frac{{(x - 5)}^{2}}{2}) = - 2 \cdot 3$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{{(x - 5)}^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{{(x - 5)}^{2}}{2}$ into the numerator.

$- 2 \frac{- {(x - 5)}^{2}}{2} = - 2 \cdot 3$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- {(x - 5)}^{2}}{2} = - 2 \cdot 3$

Cancel the common factor.

$2 \cdot - 1 \frac{- {(x - 5)}^{2}}{2} = - 2 \cdot 3$

Rewrite the expression.

$- - {(x - 5)}^{2} = - 2 \cdot 3$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 {(x - 5)}^{2} = - 2 \cdot 3$

Multiply ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = - 2 \cdot 3$

Simplify the right side.

Multiply $- 2$ by $3$ .

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

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

$x - 5 = \pm \sqrt{- 6}$

Simplify $\pm \sqrt{- 6}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x - 5 = \pm \sqrt{- 1 (6)}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x - 5 = \pm \sqrt{- 1} \cdot \sqrt{6}$

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

$x - 5 = \pm i \sqrt{6}$

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

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

$x - 5 = i \sqrt{6}$

Add $5$ to both sides of the equation.

$x = i \sqrt{6} + 5$

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

$x - 5 = - i \sqrt{6}$

Add $5$ to both sides of the equation.

$x = - i \sqrt{6} + 5$

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

$x = i \sqrt{6} + 5, - i \sqrt{6} + 5$

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

x-intercept(s): $None$

Find the y-intercepts.

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

$y = - \frac{1}{2} \cdot {((0) - 5)}^{2} - 3$

Solve the equation.

Remove parentheses.

$y = - \frac{1}{2} \cdot {(0 - 5)}^{2} - 3$

Remove parentheses.

$y = - \frac{1}{2} \cdot {((0) - 5)}^{2} - 3$

Simplify $- \frac{1}{2} \cdot {((0) - 5)}^{2} - 3$ .

Simplify each term.

Subtract $5$ from $0$ .

$y = - \frac{1}{2} \cdot {(- 5)}^{2} - 3$

Raise $- 5$ to the power of $2$ .

$y = - \frac{1}{2} \cdot 25 - 3$

Multiply $- \frac{1}{2} \cdot 25$ .

Multiply $25$ by $- 1$ .

$y = - 25 (\frac{1}{2}) - 3$

Combine $- 25$ and $\frac{1}{2}$ .

$y = \frac{- 25}{2} - 3$

Move the negative in front of the fraction.

$y = - \frac{25}{2} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{25}{2} - 3 \cdot \frac{2}{2}$

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

$y = - \frac{25}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 25 - 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$y = \frac{- 25 - 6}{2}$

Subtract $6$ from $- 25$ .

$y = \frac{- 31}{2}$

Move the negative in front of the fraction.

$y = - \frac{31}{2}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{31}{2})$

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{31}{2})$

Find the x and y Intercepts f(x)=-1/2(x-5)^2-3

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