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

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

Subtract $x$ from both sides of the equation.

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

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} + 2 - x$ .

Move $2$ .

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

Factor $- 1$ out of $- x^{2}$ .

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

Factor $- 1$ out of $- x$ .

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

Rewrite $2$ as $- 1 (- 2)$ .

$- (x^{2}) - (x) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (x^{2}) - (x)$ .

$- (x^{2} + x) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 2)$ .

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

Factor.

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

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 $- 2$ and whose sum is $1$ .

$- 1, 2 + y = x$

Write the factored form using these integers.

$- ((x - 1) (x + 2)) = 0$

Remove unnecessary parentheses.

$- (x - 1) (x + 2) = 0$

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

$x - 1 = 0$

$x + 2 = 0 + y = x$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $- (x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Step 3

Evaluate $y$ when $x = 1$ .

Substitute $1$ for $x$ .

$y = 1$

Remove parentheses.

$y = 1$

Step 4

Evaluate $y$ when $x = - 2$ .

Substitute $- 2$ for $x$ .

$y = - 2$

Remove parentheses.

$y = - 2$

Step 5

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

$(1, 1)$

$(- 2, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, 1), (- 2, - 2)$

Equation Form:

$x = 1, y = 1$

$x = - 2, y = - 2$

Step 7

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

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