Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

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

Step 1.2.1.1.2

Multiply $4$ by $0$ .

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

Step 1.2.1.1.3

Multiply $- 80$ by $0$ .

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

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

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

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

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

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

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

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

Step 1.2.4.1.2

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

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Step 1.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.4.1.2.2

Multiply $- 4$ by $320$ .

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

Step 1.2.4.1.3

Subtract $1280$ from $256$ .

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

Step 1.2.4.1.4

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

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

Step 1.2.4.1.5

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

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

Step 1.2.4.1.6

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

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

Step 1.2.4.1.7

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

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

Step 1.2.4.1.8

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

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

Step 1.2.4.1.9

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

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

Step 1.2.4.2

Multiply $2$ by $1$ .

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

Step 1.2.4.3

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

$x = - 8 \pm 16 i$

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

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

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.5.1.2.2

Multiply $- 4$ by $320$ .

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

Step 1.2.5.1.3

Subtract $1280$ from $256$ .

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

Step 1.2.5.1.4

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

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

Step 1.2.5.1.5

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

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

Step 1.2.5.1.6

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

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

Step 1.2.5.1.7

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

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

Step 1.2.5.1.8

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

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

Step 1.2.5.1.9

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

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

Step 1.2.5.2

Multiply $2$ by $1$ .

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

Step 1.2.5.3

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

$x = - 8 \pm 16 i$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = - 8 + 16 i$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

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

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

Step 1.2.6.1.2

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

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.6.1.2.2

Multiply $- 4$ by $320$ .

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

Step 1.2.6.1.3

Subtract $1280$ from $256$ .

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

Step 1.2.6.1.4

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

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

Step 1.2.6.1.5

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

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

Step 1.2.6.1.6

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

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

Step 1.2.6.1.7

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

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

Step 1.2.6.1.8

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

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

Step 1.2.6.1.9

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

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

Step 1.2.6.2

Multiply $2$ by $1$ .

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

Step 1.2.6.3

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

$x = - 8 \pm 16 i$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = - 8 - 16 i$

Step 1.2.7

The final answer is the combination of both solutions.

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

Step 1.3

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.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Multiply $16$ by $0$ .

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Factor $4$ out of $320$ .

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

Step 2.2.2.4

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

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

Step 2.2.2.5

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

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $4$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

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

Step 2.2.3.2.1.2

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

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

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $4$ .

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

Step 2.2.4

Use the quadratic formula to find the solutions.

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

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

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

Step 2.2.6.1.2

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

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.6.1.2.2

Multiply $- 4$ by $80$ .

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

Step 2.2.6.1.3

Subtract $320$ from $400$ .

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

Step 2.2.6.1.4

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

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Step 2.2.6.1.4.1

Factor $16$ out of $80$ .

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

Step 2.2.6.1.4.2

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

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

Step 2.2.6.1.5

Pull terms out from under the radical.

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

Step 2.2.6.2

Multiply $2$ by $1$ .

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

Step 2.2.6.3

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

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

Step 2.2.7

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

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

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

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

Step 2.2.7.1.2

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

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Step 2.2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.7.1.2.2

Multiply $- 4$ by $80$ .

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

Step 2.2.7.1.3

Subtract $320$ from $400$ .

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

Step 2.2.7.1.4

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

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Step 2.2.7.1.4.1

Factor $16$ out of $80$ .

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

Step 2.2.7.1.4.2

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

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

Step 2.2.7.1.5

Pull terms out from under the radical.

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

Step 2.2.7.2

Multiply $2$ by $1$ .

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

Step 2.2.7.3

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

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

Step 2.2.7.4

Change the $\pm$ to $+$ .

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

Step 2.2.8

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

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Step 2.2.8.1

Simplify the numerator.

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Step 2.2.8.1.1

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

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

Step 2.2.8.1.2

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

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Step 2.2.8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.8.1.2.2

Multiply $- 4$ by $80$ .

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

Step 2.2.8.1.3

Subtract $320$ from $400$ .

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

Step 2.2.8.1.4

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

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Step 2.2.8.1.4.1

Factor $16$ out of $80$ .

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

Step 2.2.8.1.4.2

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

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

Step 2.2.8.1.5

Pull terms out from under the radical.

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

Step 2.2.8.2

Multiply $2$ by $1$ .

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

Step 2.2.8.3

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

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

Step 2.2.8.4

Change the $\pm$ to $-$ .

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

Step 2.2.9

The final answer is the combination of both solutions.

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

Step 2.3

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