Solve the System of Equations y=-3x^2+8x+22 2=2x-y

$y = - 3 x^{2} + 8 x + 22$ $2 = 2 x - y$

Step 1

Replace all occurrences of $y$ with $- 3 x^{2} + 8 x + 22$ in each equation.

Replace all occurrences of $y$ in $2 = 2 x - y$ with $- 3 x^{2} + 8 x + 22$ .

$2 = 2 x - (- 3 x^{2} + 8 x + 22)$

$y = - 3 x^{2} + 8 x + 22$

Simplify the right side.

Simplify $2 x - (- 3 x^{2} + 8 x + 22)$ .

Simplify each term.

Apply the distributive property.

$2 = 2 x - (- 3 x^{2}) - (8 x) - 1 \cdot 22$

$y = - 3 x^{2} + 8 x + 22$

Simplify.

Multiply $- 3$ by $- 1$ .

$2 = 2 x + 3 x^{2} - (8 x) - 1 \cdot 22$

$y = - 3 x^{2} + 8 x + 22$

Multiply $8$ by $- 1$ .

$2 = 2 x + 3 x^{2} - 8 x - 1 \cdot 22$

$y = - 3 x^{2} + 8 x + 22$

Multiply $- 1$ by $22$ .

$2 = 2 x + 3 x^{2} - 8 x - 22$

$y = - 3 x^{2} + 8 x + 22$

$2 = 2 x + 3 x^{2} - 8 x - 22$

$y = - 3 x^{2} + 8 x + 22$

$2 = 2 x + 3 x^{2} - 8 x - 22$

$y = - 3 x^{2} + 8 x + 22$

Subtract $8 x$ from $2 x$ .

$2 = 3 x^{2} - 6 x - 22$

$y = - 3 x^{2} + 8 x + 22$

$2 = 3 x^{2} - 6 x - 22$

$y = - 3 x^{2} + 8 x + 22$

$2 = 3 x^{2} - 6 x - 22$

$y = - 3 x^{2} + 8 x + 22$

$2 = 3 x^{2} - 6 x - 22$

$y = - 3 x^{2} + 8 x + 22$

Step 2

Solve for $x$ in $2 = 3 x^{2} - 6 x - 22$ .

Rewrite the equation as $3 x^{2} - 6 x - 22 = 2$ .

$3 x^{2} - 6 x - 22 = 2$

$y = - 3 x^{2} + 8 x + 22$

Subtract $2$ from both sides of the equation.

$3 x^{2} - 6 x - 22 - 2 = 0$

$y = - 3 x^{2} + 8 x + 22$

Subtract $2$ from $- 22$ .

$3 x^{2} - 6 x - 24 = 0$

$y = - 3 x^{2} + 8 x + 22$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 6 x - 24$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 6 x - 24 = 0$

$y = - 3 x^{2} + 8 x + 22$

Factor $3$ out of $- 6 x$ .

$3 (x^{2}) + 3 (- 2 x) - 24 = 0$

$y = - 3 x^{2} + 8 x + 22$

Factor $3$ out of $- 24$ .

$3 x^{2} + 3 (- 2 x) + 3 \cdot - 8 = 0$

$y = - 3 x^{2} + 8 x + 22$

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

$3 (x^{2} - 2 x) + 3 \cdot - 8 = 0$

$y = - 3 x^{2} + 8 x + 22$

Factor $3$ out of $3 (x^{2} - 2 x) + 3 \cdot - 8$ .

$3 (x^{2} - 2 x - 8) = 0$

$y = - 3 x^{2} + 8 x + 22$

$3 (x^{2} - 2 x - 8) = 0$

$y = - 3 x^{2} + 8 x + 22$

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 $- 2$ .

$- 4, 2$

$y = - 3 x^{2} + 8 x + 22$

Write the factored form using these integers.

$3 ((x - 4) (x + 2)) = 0$

$y = - 3 x^{2} + 8 x + 22$

$3 ((x - 4) (x + 2)) = 0$

$y = - 3 x^{2} + 8 x + 22$

Remove unnecessary parentheses.

$3 (x - 4) (x + 2) = 0$

$y = - 3 x^{2} + 8 x + 22$

$3 (x - 4) (x + 2) = 0$

$y = - 3 x^{2} + 8 x + 22$

$3 (x - 4) (x + 2) = 0$

$y = - 3 x^{2} + 8 x + 22$

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

$x - 4 = 0$

$x + 2 = 0$

$y = - 3 x^{2} + 8 x + 22$

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

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

$x - 4 = 0$

$y = - 3 x^{2} + 8 x + 22$

Add $4$ to both sides of the equation.

$x = 4$

$y = - 3 x^{2} + 8 x + 22$

$x = 4$

$y = - 3 x^{2} + 8 x + 22$

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

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

$x + 2 = 0$

$y = - 3 x^{2} + 8 x + 22$

Subtract $2$ from both sides of the equation.

$x = - 2$

$y = - 3 x^{2} + 8 x + 22$

$x = - 2$

$y = - 3 x^{2} + 8 x + 22$

The final solution is all the values that make $3 (x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

$y = - 3 x^{2} + 8 x + 22$

$x = 4, - 2$

$y = - 3 x^{2} + 8 x + 22$

Step 3

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

Replace all occurrences of $x$ in $y = - 3 x^{2} + 8 x + 22$ with $4$ .

$y = - 3 {(4)}^{2} + 8 (4) + 22$

$x = 4$

Simplify the right side.

Simplify $- 3 {(4)}^{2} + 8 (4) + 22$ .

Simplify each term.

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

$y = - 3 \cdot 16 + 8 (4) + 22$

$x = 4$

Multiply $- 3$ by $16$ .

$y = - 48 + 8 (4) + 22$

$x = 4$

Multiply $8$ by $4$ .

$y = - 48 + 32 + 22$

$x = 4$

$y = - 48 + 32 + 22$

$x = 4$

Simplify by adding numbers.

Add $- 48$ and $32$ .

$y = - 16 + 22$

$x = 4$

Add $- 16$ and $22$ .

$y = 6$

$x = 4$

$y = 6$

$x = 4$

$y = 6$

$x = 4$

$y = 6$

$x = 4$

$y = 6$

$x = 4$

Step 4

Replace all occurrences of $x$ with $- 2$ in each equation.

Replace all occurrences of $x$ in $y = - 3 x^{2} + 8 x + 22$ with $- 2$ .

$y = - 3 {(- 2)}^{2} + 8 (- 2) + 22$

$x = - 2$

Simplify the right side.

Simplify $- 3 {(- 2)}^{2} + 8 (- 2) + 22$ .

Simplify each term.

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

$y = - 3 \cdot 4 + 8 (- 2) + 22$

$x = - 2$

Multiply $- 3$ by $4$ .

$y = - 12 + 8 (- 2) + 22$

$x = - 2$

Multiply $8$ by $- 2$ .

$y = - 12 - 16 + 22$

$x = - 2$

$y = - 12 - 16 + 22$

$x = - 2$

Simplify by adding and subtracting.

Subtract $16$ from $- 12$ .

$y = - 28 + 22$

$x = - 2$

Add $- 28$ and $22$ .

$y = - 6$

$x = - 2$

$y = - 6$

$x = - 2$

$y = - 6$

$x = - 2$

$y = - 6$

$x = - 2$

$y = - 6$

$x = - 2$

Step 5

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

$(4, 6)$

$(- 2, - 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 6), (- 2, - 6)$

Equation Form:

$x = 4, y = 6$

$x = - 2, y = - 6$

Step 7

Solve the System of Equations y=-3x^2+8x+22 2=2x-y

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