Find the x and y Intercepts 4x-x^2-13

$4 x - x^{2} - 13$

Step 1

Write $4 x - x^{2} - 13$ as an equation.

$y = 4 x - x^{2} - 13$

Step 2

Find the x-intercepts.

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

$0 = 4 x - x^{2} - 13$

Solve the equation.

Rewrite the equation as $4 x - x^{2} - 13 = 0$ .

$4 x - x^{2} - 13 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 4$ , and $c = - 13$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (- 1 \cdot - 13)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 13}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 13$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot - 13}}{2 \cdot - 1}$

Multiply $4$ by $- 13$ .

$x = \frac{- 4 \pm \sqrt{16 - 52}}{2 \cdot - 1}$

Subtract $52$ from $16$ .

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

Rewrite $- 36$ as $- 1 (36)$ .

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

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

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

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

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

Rewrite $36$ as $6^{2}$ .

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

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

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

Move $6$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 6 i}{- 2}$

Simplify $\frac{- 4 \pm 6 i}{- 2}$ .

$x = 2 \pm 3 i$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 13}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 13$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot - 13}}{2 \cdot - 1}$

Multiply $4$ by $- 13$ .

$x = \frac{- 4 \pm \sqrt{16 - 52}}{2 \cdot - 1}$

Subtract $52$ from $16$ .

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

Rewrite $- 36$ as $- 1 (36)$ .

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

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

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

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

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

Rewrite $36$ as $6^{2}$ .

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

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

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

Move $6$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 6 i}{- 2}$

Simplify $\frac{- 4 \pm 6 i}{- 2}$ .

$x = 2 \pm 3 i$

Change the $\pm$ to $+$ .

$x = 2 + 3 i$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 13}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 13$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot - 13}}{2 \cdot - 1}$

Multiply $4$ by $- 13$ .

$x = \frac{- 4 \pm \sqrt{16 - 52}}{2 \cdot - 1}$

Subtract $52$ from $16$ .

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

Rewrite $- 36$ as $- 1 (36)$ .

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

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

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

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

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

Rewrite $36$ as $6^{2}$ .

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

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

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

Move $6$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 6 i}{- 2}$

Simplify $\frac{- 4 \pm 6 i}{- 2}$ .

$x = 2 \pm 3 i$

Change the $\pm$ to $-$ .

$x = 2 - 3 i$

The final answer is the combination of both solutions.

$x = 2 + 3 i, 2 - 3 i$

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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

$y = 4 (0) - {(0)}^{2} - 13$

Solve the equation.

Remove parentheses.

$y = 4 (0) - 0^{2} - 13$

Remove parentheses.

$y = 4 (0) - {(0)}^{2} - 13$

Simplify $4 (0) - {(0)}^{2} - 13$ .

Simplify each term.

Multiply $4$ by $0$ .

$y = 0 - {(0)}^{2} - 13$

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

$y = 0 - 0 - 13$

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 13$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 13$

Subtract $13$ from $0$ .

$y = - 13$

y-intercept(s) in point form.

y-intercept(s): $(0, - 13)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 13)$

Step 5

Find the x and y Intercepts 4x-x^2-13

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