Solve the System of Equations y=x^2 and y=x-2

$y = x^{2}$ and $y = x - 2$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} = x - 2$

Step 2

Solve $x^{2} = x - 2$ for $x$ .

Subtract $x$ from both sides of the equation.

$x^{2} - x = - 2$

Add $2$ to both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1} + y = x - 2$

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Subtract $8$ from $1$ .

$x = \frac{1 \pm \sqrt{- 7}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{7}}{2}$

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

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Subtract $8$ from $1$ .

$x = \frac{1 \pm \sqrt{- 7}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{7}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{7}}{2}$

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

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Subtract $8$ from $1$ .

$x = \frac{1 \pm \sqrt{- 7}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{7}}{2}$

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

Step 3

Evaluate $y$ when $x = \frac{1 + i \sqrt{7}}{2}$ .

Substitute $\frac{1 + i \sqrt{7}}{2}$ for $x$ .

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

Substitute $\frac{1 + i \sqrt{7}}{2}$ for $x$ in $y = (\frac{1 + i \sqrt{7}}{2}) - 2$ and solve for $y$ .

Remove parentheses.

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

Remove parentheses.

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

Simplify $(\frac{1 + i \sqrt{7}}{2}) - 2$ .

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

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

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

Move the negative in front of the fraction.

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

Step 4

Evaluate $y$ when $x = \frac{1 - i \sqrt{7}}{2}$ .

Substitute $\frac{1 - i \sqrt{7}}{2}$ for $x$ .

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

Substitute $\frac{1 - i \sqrt{7}}{2}$ for $x$ in $y = (\frac{1 - i \sqrt{7}}{2}) - 2$ and solve for $y$ .

Remove parentheses.

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

Remove parentheses.

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

Simplify $(\frac{1 - i \sqrt{7}}{2}) - 2$ .

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

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

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

Move the negative in front of the fraction.

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

Step 5

List all of the solutions.

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

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

Step 6

Solve the System of Equations y=x^2 and y=x-2

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