Precalculus Examples

$x^{2} + y = 0$ , $x^{3} - 12 x - y = 0$

Step 1

Subtract $x^{2}$ from both sides of the equation.

$y = - x^{2}$

$x^{3} - 12 x - y = 0$

Step 2

Replace all occurrences of $y$ with $- x^{2}$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $y$ in $x^{3} - 12 x - y = 0$ with $- x^{2}$ .

$x^{3} - 12 x - (- x^{2}) = 0$

$y = - x^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Multiply $- (- x^{2})$ .

Tap for more steps...

Step 2.2.1.1

Multiply $- 1$ by $- 1$ .

$x^{3} - 12 x + 1 x^{2} = 0$

$y = - x^{2}$

Step 2.2.1.2

Multiply $x^{2}$ by $1$ .

$x^{3} - 12 x + x^{2} = 0$

$y = - x^{2}$

$x^{3} - 12 x + x^{2} = 0$

$y = - x^{2}$

$x^{3} - 12 x + x^{2} = 0$

$y = - x^{2}$

$x^{3} - 12 x + x^{2} = 0$

$y = - x^{2}$

Step 3

Solve for $x$ in $x^{3} - 12 x + x^{2} = 0$ .

Tap for more steps...

Step 3.1

Factor the left side of the equation.

Tap for more steps...

Step 3.1.1

Factor $x$ out of $x^{3} - 12 x + x^{2}$ .

Tap for more steps...

Step 3.1.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 12 x + x^{2} = 0$

$y = - x^{2}$

Step 3.1.1.2

Factor $x$ out of $- 12 x$ .

$x \cdot x^{2} + x \cdot - 12 + x^{2} = 0$

$y = - x^{2}$

Step 3.1.1.3

Factor $x$ out of $x^{2}$ .

$x \cdot x^{2} + x \cdot - 12 + x \cdot x = 0$

$y = - x^{2}$

Step 3.1.1.4

Factor $x$ out of $x \cdot x^{2} + x \cdot - 12$ .

$x (x^{2} - 12) + x \cdot x = 0$

$y = - x^{2}$

Step 3.1.1.5

Factor $x$ out of $x (x^{2} - 12) + x \cdot x$ .

$x (x^{2} - 12 + x) = 0$

$y = - x^{2}$

$x (x^{2} - 12 + x) = 0$

$y = - x^{2}$

Step 3.1.2

Factor.

Tap for more steps...

Step 3.1.2.1

Factor $x^{2} - 12 + x$ using the AC method.

Tap for more steps...

Step 3.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

$y = - x^{2}$

Step 3.1.2.1.2

Write the factored form using these integers.

$x ((x - 3) (x + 4)) = 0$

$y = - x^{2}$

$x ((x - 3) (x + 4)) = 0$

$y = - x^{2}$

Step 3.1.2.2

Remove unnecessary parentheses.

$x (x - 3) (x + 4) = 0$

$y = - x^{2}$

$x (x - 3) (x + 4) = 0$

$y = - x^{2}$

$x (x - 3) (x + 4) = 0$

$y = - x^{2}$

Step 3.2

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

$x = 0$

$x - 3 = 0$

$x + 4 = 0$

$y = - x^{2}$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

$y = - x^{2}$

Step 3.4

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

$y = - x^{2}$

Step 3.4.2

Add $3$ to both sides of the equation.

$x = 3$

$y = - x^{2}$

$x = 3$

$y = - x^{2}$

Step 3.5

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

$y = - x^{2}$

Step 3.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

$y = - x^{2}$

$x = - 4$

$y = - x^{2}$

Step 3.6

The final solution is all the values that make $x (x - 3) (x + 4) = 0$ true.

$x = 0, 3, - 4$

$y = - x^{2}$

$x = 0, 3, - 4$

$y = - x^{2}$

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = - x^{2}$ with $0$ .

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

$x = 0$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

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

Tap for more steps...

Step 4.2.1.1

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

$y = - 0$

$x = 0$

Step 4.2.1.2

Multiply $- 1$ by $0$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 5

Replace all occurrences of $x$ with $3$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $x$ in $y = - x^{2}$ with $3$ .

$y = - {(3)}^{2}$

$x = 3$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $- {(3)}^{2}$ .

Tap for more steps...

Step 5.2.1.1

Raise $3$ to the power of $2$ .

$y = - 1 \cdot 9$

$x = 3$

Step 5.2.1.2

Multiply $- 1$ by $9$ .

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

Step 6

Replace all occurrences of $x$ with $0$ in each equation.

Tap for more steps...

Step 6.1

Replace all occurrences of $x$ in $y = - x^{2}$ with $0$ .

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

$x = 0$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

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

Tap for more steps...

Step 6.2.1.1

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

$y = - 0$

$x = 0$

Step 6.2.1.2

Multiply $- 1$ by $0$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 7

Replace all occurrences of $x$ with $3$ in each equation.

Tap for more steps...

Step 7.1

Replace all occurrences of $x$ in $y = - x^{2}$ with $3$ .

$y = - {(3)}^{2}$

$x = 3$

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

Simplify $- {(3)}^{2}$ .

Tap for more steps...

Step 7.2.1.1

Raise $3$ to the power of $2$ .

$y = - 1 \cdot 9$

$x = 3$

Step 7.2.1.2

Multiply $- 1$ by $9$ .

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

$y = - 9$

$x = 3$

Step 8

Replace all occurrences of $x$ with $- 4$ in each equation.

Tap for more steps...

Step 8.1

Replace all occurrences of $x$ in $y = - x^{2}$ with $- 4$ .

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

$x = - 4$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

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

Tap for more steps...

Step 8.2.1.1

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

$y = - 1 \cdot 16$

$x = - 4$

Step 8.2.1.2

Multiply $- 1$ by $16$ .

$y = - 16$

$x = - 4$

$y = - 16$

$x = - 4$

$y = - 16$

$x = - 4$

$y = - 16$

$x = - 4$

Step 9

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

$(3, - 9)$

$(- 4, - 16)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(0, 0), (3, - 9), (- 4, - 16)$

Equation Form:

$x = 0, y = 0$

$x = 3, y = - 9$

$x = - 4, y = - 16$

Step 11