Solve by Substitution y=-1/2x^2 y=x-4

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $- \frac{1}{2} x^{2} = x - 4$ for $x$ .

Simplify $- \frac{1}{2} x^{2}$ .

Rewrite.

$0 + 0 - \frac{1}{2} x^{2} = x - 4$

Simplify by adding zeros.

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

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

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

Subtract $x$ from both sides of the equation.

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

Multiply each term in $- \frac{x^{2}}{2} - x = - 4$ by $2$ to eliminate the fractions.

Multiply each term in $- \frac{x^{2}}{2} - x = - 4$ by $2$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 1$ .

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

Simplify the right side.

Multiply $- 4$ by $2$ .

$- x^{2} - 2 x = - 8$

Add $8$ to both sides of the equation.

$- x^{2} - 2 x + 8 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 2 x + 8$ .

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

$- (x^{2}) - 2 x + 8 = 0$

Factor $- 1$ out of $- 2 x$ .

$- (x^{2}) - (2 x) + 8 = 0$

Rewrite $8$ as $- 1 (- 8)$ .

$- (x^{2}) - (2 x) - 1 \cdot - 8 = 0$

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

$- (x^{2} + 2 x) - 1 \cdot - 8 = 0$

Factor $- 1$ out of $- (x^{2} + 2 x) - 1 (- 8)$ .

$- (x^{2} + 2 x - 8) = 0$

Factor.

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4 + y = x - 4$

Write the factored form using these integers.

$- ((x - 2) (x + 4)) = 0$

Remove unnecessary parentheses.

$- (x - 2) (x + 4) = 0$

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

$x - 2 = 0$

$x + 4 = 0 + y = x - 4$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $- (x - 2) (x + 4) = 0$ true.

$x = 2, - 4$

Step 3

Evaluate $y$ when $x = 2$ .

Substitute $2$ for $x$ .

$y = (2) - 4$

Substitute $2$ for $x$ in $y = (2) - 4$ and solve for $y$ .

Remove parentheses.

$y = 2 - 4$

Remove parentheses.

$y = (2) - 4$

Subtract $4$ from $2$ .

$y = - 2$

Step 4

Evaluate $y$ when $x = - 4$ .

Substitute $- 4$ for $x$ .

$y = (- 4) - 4$

Substitute $- 4$ for $x$ in $y = (- 4) - 4$ and solve for $y$ .

Remove parentheses.

$y = - 4 - 4$

Remove parentheses.

$y = (- 4) - 4$

Subtract $4$ from $- 4$ .

$y = - 8$

Step 5

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

$(2, - 2)$

$(- 4, - 8)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 2), (- 4, - 8)$

Equation Form:

$x = 2, y = - 2$

$x = - 4, y = - 8$

Step 7

Solve by Substitution y=-1/2x^2 y=x-4

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