Algebra Examples

$x^{2} + y^{3} - x^{2} y^{2} = 64$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$x^{2} + {(0)}^{3} - x^{2} {(0)}^{2} = 64$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Simplify $x^{2} + {(0)}^{3} - x^{2} {(0)}^{2}$ .

Tap for more steps...

Step 1.2.1.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1.1

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

$x^{2} + 0 - x^{2} {(0)}^{2} = 64$

Step 1.2.1.1.2

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

$x^{2} + 0 - x^{2} \cdot 0 = 64$

Step 1.2.1.1.3

Multiply $- x^{2} \cdot 0$ .

Tap for more steps...

Step 1.2.1.1.3.1

Multiply $0$ by $- 1$ .

$x^{2} + 0 + 0 x^{2} = 64$

Step 1.2.1.1.3.2

Multiply $0$ by $x^{2}$ .

$x^{2} + 0 + 0 = 64$

Step 1.2.1.2

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

Tap for more steps...

Step 1.2.1.2.1

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

$x^{2} + 0 = 64$

Step 1.2.1.2.2

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

$x^{2} = 64$

Step 1.2.2

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

$x = \pm \sqrt{64}$

Step 1.2.3

Simplify $\pm \sqrt{64}$ .

Tap for more steps...

Step 1.2.3.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 1.2.3.2

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

$x = \pm 8$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$x = 8$

Step 1.2.4.2

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

$x = - 8$

Step 1.2.4.3

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

$x = 8, - 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(8, 0), (- 8, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

${(0)}^{2} + y^{3} - {(0)}^{2} y^{2} = 64$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Simplify ${(0)}^{2} + y^{3} - {(0)}^{2} y^{2}$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

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

$0 + y^{3} - {(0)}^{2} y^{2} = 64$

Step 2.2.1.1.2

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

$0 + y^{3} - 0 y^{2} = 64$

Step 2.2.1.1.3

Multiply $- 1$ by $0$ .

$0 + y^{3} + 0 y^{2} = 64$

Step 2.2.1.1.4

Multiply $0$ by $y^{2}$ .

$0 + y^{3} + 0 = 64$

Step 2.2.1.2

Combine the opposite terms in $0 + y^{3} + 0$ .

Tap for more steps...

Step 2.2.1.2.1

Add $0$ and $y^{3}$ .

$y^{3} + 0 = 64$

Step 2.2.1.2.2

Add $y^{3}$ and $0$ .

$y^{3} = 64$

Step 2.2.2

Subtract $64$ from both sides of the equation.

$y^{3} - 64 = 0$

Step 2.2.3

Factor the left side of the equation.

Tap for more steps...

Step 2.2.3.1

Rewrite $64$ as $4^{3}$ .

$y^{3} - 4^{3} = 0$

Step 2.2.3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 4$ .

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

Step 2.2.3.3

Simplify.

Tap for more steps...

Step 2.2.3.3.1

Move $4$ to the left of $y$ .

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

Step 2.2.3.3.2

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

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

Step 2.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 4 = 0$

$y^{2} + 4 y + 16 = 0$

Step 2.2.5

Set $y - 4$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.2.5.1

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

Step 2.2.5.2

Add $4$ to both sides of the equation.

$y = 4$

Step 2.2.6

Set $y^{2} + 4 y + 16$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.2.6.1

Set $y^{2} + 4 y + 16$ equal to $0$ .

$y^{2} + 4 y + 16 = 0$

Step 2.2.6.2

Solve $y^{2} + 4 y + 16 = 0$ for $y$ .

Tap for more steps...

Step 2.2.6.2.1

Use the quadratic formula to find the solutions.

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

Step 2.2.6.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $y$ .

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

Step 2.2.6.2.3

Simplify.

Tap for more steps...

Step 2.2.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.2.6.2.3.1.1

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

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

Step 2.2.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 2.2.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.6.2.3.1.2.2

Multiply $- 4$ by $16$ .

$y = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 2.2.6.2.3.1.3

Subtract $64$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 2.2.6.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 2.2.6.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.3.1.6

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

$y = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 2.2.6.2.3.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.2.6.2.3.1.7.2

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

$y = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.2.6.2.3.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 2.2.6.2.3.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 2.2.6.2.3.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 2.2.6.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$y = - 2 \pm 2 i \sqrt{3}$

Step 2.2.6.2.4

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

Tap for more steps...

Step 2.2.6.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.6.2.4.1.1

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

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

Step 2.2.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 2.2.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.6.2.4.1.2.2

Multiply $- 4$ by $16$ .

$y = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 2.2.6.2.4.1.3

Subtract $64$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 2.2.6.2.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 2.2.6.2.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.4.1.6

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

$y = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 2.2.6.2.4.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.2.6.2.4.1.7.2

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

$y = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.2.6.2.4.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 2.2.6.2.4.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 2.2.6.2.4.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 2.2.6.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$y = - 2 \pm 2 i \sqrt{3}$

Step 2.2.6.2.4.4

Change the $\pm$ to $+$ .

$y = - 2 + 2 i \sqrt{3}$

Step 2.2.6.2.5

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

Tap for more steps...

Step 2.2.6.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.2.6.2.5.1.1

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

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

Step 2.2.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 2.2.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.6.2.5.1.2.2

Multiply $- 4$ by $16$ .

$y = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 2.2.6.2.5.1.3

Subtract $64$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 2.2.6.2.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 2.2.6.2.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.5.1.6

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

$y = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.2.6.2.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 2.2.6.2.5.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.2.6.2.5.1.7.2

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

$y = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.2.6.2.5.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 2.2.6.2.5.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 2.2.6.2.5.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 2.2.6.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$y = - 2 \pm 2 i \sqrt{3}$

Step 2.2.6.2.5.4

Change the $\pm$ to $-$ .

$y = - 2 - 2 i \sqrt{3}$

Step 2.2.6.2.6

The final answer is the combination of both solutions.

$y = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 2.2.7

The final solution is all the values that make $(y - 4) (y^{2} + 4 y + 16) = 0$ true.

$y = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $(8, 0), (- 8, 0)$

y-intercept(s): $(0, 4)$

Step 4