Algebra Examples

$x^{2} - 4 y = 12$ $y = - \frac{1}{2} x - 4$

Step 1

Combine $x$ and $\frac{1}{2}$ .

$y = - \frac{x}{2} - 4$

$x^{2} - 4 y = 12$

Step 2

Replace all occurrences of $y$ with $- \frac{x}{2} - 4$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $y$ in $x^{2} - 4 y = 12$ with $- \frac{x}{2} - 4$ .

$x^{2} - 4 (- \frac{x}{2} - 4) = 12$

$y = - \frac{x}{2} - 4$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$x^{2} - 4 (- \frac{x}{2}) - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.2.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$x^{2} - 4 \frac{- x}{2} - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.2.2

Factor $2$ out of $- 4$ .

$x^{2} + 2 (- 2) (\frac{- x}{2}) - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.2.3

Cancel the common factor.

$x^{2} + 2 \cdot (- 2 \frac{- x}{2}) - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.2.4

Rewrite the expression.

$x^{2} - 2 (- x) - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

$x^{2} - 2 (- x) - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.3

Multiply $- 1$ by $- 2$ .

$x^{2} + 2 x - 4 \cdot - 4 = 12$

$y = - \frac{x}{2} - 4$

Step 2.2.1.4

Multiply $- 4$ by $- 4$ .

$x^{2} + 2 x + 16 = 12$

$y = - \frac{x}{2} - 4$

$x^{2} + 2 x + 16 = 12$

$y = - \frac{x}{2} - 4$

$x^{2} + 2 x + 16 = 12$

$y = - \frac{x}{2} - 4$

$x^{2} + 2 x + 16 = 12$

$y = - \frac{x}{2} - 4$

Step 3

Solve for $x$ in $x^{2} + 2 x + 16 = 12$ .

Tap for more steps...

Step 3.1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 3.1.1

Subtract $12$ from both sides of the equation.

$x^{2} + 2 x + 16 - 12 = 0$

$y = - \frac{x}{2} - 4$

Step 3.1.2

Subtract $12$ from $16$ .

$x^{2} + 2 x + 4 = 0$

$y = - \frac{x}{2} - 4$

$x^{2} + 2 x + 4 = 0$

$y = - \frac{x}{2} - 4$

Step 3.2

Use the quadratic formula to find the solutions.

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

$y = - \frac{x}{2} - 4$

Step 3.3

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

$y = - \frac{x}{2} - 4$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Simplify the numerator.

Tap for more steps...

Step 3.4.1.1

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

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.2

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

Tap for more steps...

Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.2.2

Multiply $- 4$ by $4$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 3.4.1.7.1

Factor $4$ out of $12$ .

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.7.2

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

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.8

Pull terms out from under the radical.

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

$y = - \frac{x}{2} - 4$

Step 3.4.1.9

Move $2$ to the left of $i$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.4.2

Multiply $2$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.4.3

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

$x = - 1 \pm i \sqrt{3}$

$y = - \frac{x}{2} - 4$

$x = - 1 \pm i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 3.5

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

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

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

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.2

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

Tap for more steps...

Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.2.2

Multiply $- 4$ by $4$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.5.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 3.5.1.7.1

Factor $4$ out of $12$ .

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.7.2

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

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.8

Pull terms out from under the radical.

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

$y = - \frac{x}{2} - 4$

Step 3.5.1.9

Move $2$ to the left of $i$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.5.2

Multiply $2$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.5.3

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

$x = - 1 \pm i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 3.5.4

Change the $\pm$ to $+$ .

$x = - 1 + i \sqrt{3}$

$y = - \frac{x}{2} - 4$

$x = - 1 + i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 3.6

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

Tap for more steps...

Step 3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.6.1.1

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

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.2

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

Tap for more steps...

Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.2.2

Multiply $- 4$ by $4$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.6.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.6.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$y = - \frac{x}{2} - 4$

Step 3.6.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 3.6.1.7.1

Factor $4$ out of $12$ .

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.7.2

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

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.8

Pull terms out from under the radical.

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

$y = - \frac{x}{2} - 4$

Step 3.6.1.9

Move $2$ to the left of $i$ .

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

$y = - \frac{x}{2} - 4$

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

$y = - \frac{x}{2} - 4$

Step 3.6.2

Multiply $2$ by $1$ .

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

$y = - \frac{x}{2} - 4$

Step 3.6.3

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

$x = - 1 \pm i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 3.6.4

Change the $\pm$ to $-$ .

$x = - 1 - i \sqrt{3}$

$y = - \frac{x}{2} - 4$

$x = - 1 - i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 3.7

The final answer is the combination of both solutions.

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$y = - \frac{x}{2} - 4$

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$y = - \frac{x}{2} - 4$

Step 4

Replace all occurrences of $x$ with $- 1 + i \sqrt{3}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = - \frac{x}{2} - 4$ with $- 1 + i \sqrt{3}$ .

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

$x = - 1 + i \sqrt{3}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- \frac{- 1 + i \sqrt{3}}{2} - 4$ .

Tap for more steps...

Step 4.2.1.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$x = - 1 + i \sqrt{3}$

Step 4.2.1.2

Combine $- 4$ and $\frac{2}{2}$ .

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

$x = - 1 + i \sqrt{3}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$y = \frac{- 7 - i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

Step 4.2.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 - i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

Step 4.2.1.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$y = \frac{- 1 \cdot 7 - (i \sqrt{3})}{2}$

$x = - 1 + i \sqrt{3}$

Step 4.2.1.6

Factor $- 1$ out of $- 1 (7) - (i \sqrt{3})$ .

$y = \frac{- 1 (7 + i \sqrt{3})}{2}$

$x = - 1 + i \sqrt{3}$

Step 4.2.1.7

Move the negative in front of the fraction.

$y = - \frac{7 + i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

$y = - \frac{7 + i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

$y = - \frac{7 + i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

$y = - \frac{7 + i \sqrt{3}}{2}$

$x = - 1 + i \sqrt{3}$

Step 5

Replace all occurrences of $x$ with $- 1 - i \sqrt{3}$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $x$ in $y = - \frac{x}{2} - 4$ with $- 1 - i \sqrt{3}$ .

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

$x = - 1 - i \sqrt{3}$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $- \frac{- 1 - i \sqrt{3}}{2} - 4$ .

Tap for more steps...

Step 5.2.1.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$x = - 1 - i \sqrt{3}$

Step 5.2.1.2

Combine $- 4$ and $\frac{2}{2}$ .

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

$x = - 1 - i \sqrt{3}$

Step 5.2.1.3

Combine the numerators over the common denominator.

$y = \frac{- 7 + \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

Step 5.2.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 + \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

Step 5.2.1.5

Factor $- 1$ out of $\sqrt{3} i$ .

$y = \frac{- 1 \cdot 7 - (- \sqrt{3} i)}{2}$

$x = - 1 - i \sqrt{3}$

Step 5.2.1.6

Factor $- 1$ out of $- 1 (7) - (- \sqrt{3} i)$ .

$y = \frac{- 1 (7 - \sqrt{3} i)}{2}$

$x = - 1 - i \sqrt{3}$

Step 5.2.1.7

Move the negative in front of the fraction.

$y = - \frac{7 - \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

$y = - \frac{7 - \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

$y = - \frac{7 - \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

$y = - \frac{7 - \sqrt{3} i}{2}$

$x = - 1 - i \sqrt{3}$

Step 6

List all of the solutions.

$y = - \frac{7 + i \sqrt{3}}{2}, x = - 1 + i \sqrt{3}$

$y = - \frac{7 - \sqrt{3} i}{2}, x = - 1 - i \sqrt{3}$

Step 7