Find the x and y Intercepts x^2+4y^2+16x-80y+320=0

$x^{2} + 4 y^{2} + 16 x - 80 y + 320 = 0$

Step 1

Find the x-intercepts.

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

$x^{2} + 4 {(0)}^{2} + 16 x - 80 \cdot 0 + 320 = 0$

Solve the equation.

Simplify $x^{2} + 4 {(0)}^{2} + 16 x - 80 \cdot 0 + 320$ .

Simplify each term.

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

$x^{2} + 4 \cdot 0 + 16 x - 80 \cdot 0 + 320 = 0$

Multiply $4$ by $0$ .

$x^{2} + 0 + 16 x - 80 \cdot 0 + 320 = 0$

Multiply $- 80$ by $0$ .

$x^{2} + 0 + 16 x + 0 + 320 = 0$

Combine the opposite terms in $x^{2} + 0 + 16 x + 0 + 320$ .

Add $x^{2}$ and $0$ .

$x^{2} + 16 x + 0 + 320 = 0$

Add $x^{2} + 16 x$ and $0$ .

$x^{2} + 16 x + 320 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 16$ , and $c = 320$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot 320)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 320}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 320$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 320}}{2 \cdot 1}$

Multiply $- 4$ by $320$ .

$x = \frac{- 16 \pm \sqrt{256 - 1280}}{2 \cdot 1}$

Subtract $1280$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 1024}}{2 \cdot 1}$

Rewrite $- 1024$ as $- 1 (1024)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 1024}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (1024)}$ as $\sqrt{- 1} \cdot \sqrt{1024}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{1024}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot \sqrt{1024}}{2 \cdot 1}$

Rewrite $1024$ as $32^{2}$ .

$x = \frac{- 16 \pm i \cdot \sqrt{32^{2}}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot 32}{2 \cdot 1}$

Move $32$ to the left of $i$ .

$x = \frac{- 16 \pm 32 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 32 i}{2}$

Simplify $\frac{- 16 \pm 32 i}{2}$ .

$x = - 8 \pm 16 i$

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

Simplify the numerator.

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 320}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 320$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 320}}{2 \cdot 1}$

Multiply $- 4$ by $320$ .

$x = \frac{- 16 \pm \sqrt{256 - 1280}}{2 \cdot 1}$

Subtract $1280$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 1024}}{2 \cdot 1}$

Rewrite $- 1024$ as $- 1 (1024)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 1024}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (1024)}$ as $\sqrt{- 1} \cdot \sqrt{1024}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{1024}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot \sqrt{1024}}{2 \cdot 1}$

Rewrite $1024$ as $32^{2}$ .

$x = \frac{- 16 \pm i \cdot \sqrt{32^{2}}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot 32}{2 \cdot 1}$

Move $32$ to the left of $i$ .

$x = \frac{- 16 \pm 32 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 32 i}{2}$

Simplify $\frac{- 16 \pm 32 i}{2}$ .

$x = - 8 \pm 16 i$

Change the $\pm$ to $+$ .

$x = - 8 + 16 i$

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

Simplify the numerator.

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 320}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 320$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 320}}{2 \cdot 1}$

Multiply $- 4$ by $320$ .

$x = \frac{- 16 \pm \sqrt{256 - 1280}}{2 \cdot 1}$

Subtract $1280$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 1024}}{2 \cdot 1}$

Rewrite $- 1024$ as $- 1 (1024)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 1024}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (1024)}$ as $\sqrt{- 1} \cdot \sqrt{1024}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{1024}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot \sqrt{1024}}{2 \cdot 1}$

Rewrite $1024$ as $32^{2}$ .

$x = \frac{- 16 \pm i \cdot \sqrt{32^{2}}}{2 \cdot 1}$

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

$x = \frac{- 16 \pm i \cdot 32}{2 \cdot 1}$

Move $32$ to the left of $i$ .

$x = \frac{- 16 \pm 32 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 32 i}{2}$

Simplify $\frac{- 16 \pm 32 i}{2}$ .

$x = - 8 \pm 16 i$

Change the $\pm$ to $-$ .

$x = - 8 - 16 i$

The final answer is the combination of both solutions.

$x = - 8 + 16 i, - 8 - 16 i$

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.

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

${(0)}^{2} + 4 y^{2} + 16 (0) - 80 y + 320 = 0$

Solve the equation.

Simplify ${(0)}^{2} + 4 y^{2} + 16 (0) - 80 y + 320$ .

Simplify each term.

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

$0 + 4 y^{2} + 16 (0) - 80 y + 320 = 0$

Multiply $16$ by $0$ .

$0 + 4 y^{2} + 0 - 80 y + 320 = 0$

Combine the opposite terms in $0 + 4 y^{2} + 0 - 80 y + 320$ .

Add $0$ and $4 y^{2}$ .

$4 y^{2} + 0 - 80 y + 320 = 0$

Add $4 y^{2}$ and $0$ .

$4 y^{2} - 80 y + 320 = 0$

Factor $4$ out of $4 y^{2} - 80 y + 320$ .

Factor $4$ out of $4 y^{2}$ .

$4 (y^{2}) - 80 y + 320 = 0$

Factor $4$ out of $- 80 y$ .

$4 (y^{2}) + 4 (- 20 y) + 320 = 0$

Factor $4$ out of $320$ .

$4 y^{2} + 4 (- 20 y) + 4 \cdot 80 = 0$

Factor $4$ out of $4 y^{2} + 4 (- 20 y)$ .

$4 (y^{2} - 20 y) + 4 \cdot 80 = 0$

Factor $4$ out of $4 (y^{2} - 20 y) + 4 \cdot 80$ .

$4 (y^{2} - 20 y + 80) = 0$

Divide each term in $4 (y^{2} - 20 y + 80) = 0$ by $4$ and simplify.

Divide each term in $4 (y^{2} - 20 y + 80) = 0$ by $4$ .

$\frac{4 (y^{2} - 20 y + 80)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (y^{2} - 20 y + 80)}{4} = \frac{0}{4}$

Divide $y^{2} - 20 y + 80$ by $1$ .

$y^{2} - 20 y + 80 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$y^{2} - 20 y + 80 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 20$ , and $c = 80$ into the quadratic formula and solve for $y$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (1 \cdot 80)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 20$ to the power of $2$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 80}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 80$ .

Multiply $- 4$ by $1$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 80}}{2 \cdot 1}$

Multiply $- 4$ by $80$ .

$y = \frac{20 \pm \sqrt{400 - 320}}{2 \cdot 1}$

Subtract $320$ from $400$ .

$y = \frac{20 \pm \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$y = \frac{20 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{20 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{20 \pm 4 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{20 \pm 4 \sqrt{5}}{2}$

Simplify $\frac{20 \pm 4 \sqrt{5}}{2}$ .

$y = 10 \pm 2 \sqrt{5}$

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

Simplify the numerator.

Raise $- 20$ to the power of $2$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 80}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 80$ .

Multiply $- 4$ by $1$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 80}}{2 \cdot 1}$

Multiply $- 4$ by $80$ .

$y = \frac{20 \pm \sqrt{400 - 320}}{2 \cdot 1}$

Subtract $320$ from $400$ .

$y = \frac{20 \pm \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$y = \frac{20 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{20 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{20 \pm 4 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{20 \pm 4 \sqrt{5}}{2}$

Simplify $\frac{20 \pm 4 \sqrt{5}}{2}$ .

$y = 10 \pm 2 \sqrt{5}$

Change the $\pm$ to $+$ .

$y = 10 + 2 \sqrt{5}$

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

Simplify the numerator.

Raise $- 20$ to the power of $2$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 80}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 80$ .

Multiply $- 4$ by $1$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot 80}}{2 \cdot 1}$

Multiply $- 4$ by $80$ .

$y = \frac{20 \pm \sqrt{400 - 320}}{2 \cdot 1}$

Subtract $320$ from $400$ .

$y = \frac{20 \pm \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$y = \frac{20 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{20 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{20 \pm 4 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{20 \pm 4 \sqrt{5}}{2}$

Simplify $\frac{20 \pm 4 \sqrt{5}}{2}$ .

$y = 10 \pm 2 \sqrt{5}$

Change the $\pm$ to $-$ .

$y = 10 - 2 \sqrt{5}$

The final answer is the combination of both solutions.

$y = 10 + 2 \sqrt{5}, 10 - 2 \sqrt{5}$

y-intercept(s) in point form.

y-intercept(s): $(0, 10 + 2 \sqrt{5}), (0, 10 - 2 \sqrt{5})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 10 + 2 \sqrt{5}), (0, 10 - 2 \sqrt{5})$

Step 4

Find the x and y Intercepts x^2+4y^2+16x-80y+320=0

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