Solve the System of Equations y=3x^2+x-206 2x+y=10

$y = 3 x^{2} + x - 206$ $2 x + y = 10$

Step 1

Replace all occurrences of $y$ with $3 x^{2} + x - 206$ in each equation.

Replace all occurrences of $y$ in $2 x + y = 10$ with $3 x^{2} + x - 206$ .

$2 x + 3 x^{2} + x - 206 = 10$

$y = 3 x^{2} + x - 206$

Simplify the left side.

Simplify $2 x + 3 x^{2} + x - 206$ .

Remove parentheses.

$2 x + 3 x^{2} + x - 206 = 10$

$y = 3 x^{2} + x - 206$

Add $2 x$ and $x$ .

$3 x^{2} + 3 x - 206 = 10$

$y = 3 x^{2} + x - 206$

$3 x^{2} + 3 x - 206 = 10$

$y = 3 x^{2} + x - 206$

$3 x^{2} + 3 x - 206 = 10$

$y = 3 x^{2} + x - 206$

$3 x^{2} + 3 x - 206 = 10$

$y = 3 x^{2} + x - 206$

Step 2

Solve for $x$ in $3 x^{2} + 3 x - 206 = 10$ .

Subtract $10$ from both sides of the equation.

$3 x^{2} + 3 x - 206 - 10 = 0$

$y = 3 x^{2} + x - 206$

Subtract $10$ from $- 206$ .

$3 x^{2} + 3 x - 216 = 0$

$y = 3 x^{2} + x - 206$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} + 3 x - 216$ .

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

$3 (x^{2}) + 3 x - 216 = 0$

$y = 3 x^{2} + x - 206$

Factor $3$ out of $3 x$ .

$3 (x^{2}) + 3 (x) - 216 = 0$

$y = 3 x^{2} + x - 206$

Factor $3$ out of $- 216$ .

$3 (x^{2}) + 3 x + 3 \cdot - 72 = 0$

$y = 3 x^{2} + x - 206$

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

$3 (x^{2} + x) + 3 \cdot - 72 = 0$

$y = 3 x^{2} + x - 206$

Factor $3$ out of $3 (x^{2} + x) + 3 \cdot - 72$ .

$3 (x^{2} + x - 72) = 0$

$y = 3 x^{2} + x - 206$

$3 (x^{2} + x - 72) = 0$

$y = 3 x^{2} + x - 206$

Factor.

Factor $x^{2} + x - 72$ 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 $- 72$ and whose sum is $1$ .

$- 8, 9$

$y = 3 x^{2} + x - 206$

Write the factored form using these integers.

$3 ((x - 8) (x + 9)) = 0$

$y = 3 x^{2} + x - 206$

$3 ((x - 8) (x + 9)) = 0$

$y = 3 x^{2} + x - 206$

Remove unnecessary parentheses.

$3 (x - 8) (x + 9) = 0$

$y = 3 x^{2} + x - 206$

$3 (x - 8) (x + 9) = 0$

$y = 3 x^{2} + x - 206$

$3 (x - 8) (x + 9) = 0$

$y = 3 x^{2} + x - 206$

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

$x - 8 = 0$

$x + 9 = 0$

$y = 3 x^{2} + x - 206$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

$y = 3 x^{2} + x - 206$

Add $8$ to both sides of the equation.

$x = 8$

$y = 3 x^{2} + x - 206$

$x = 8$

$y = 3 x^{2} + x - 206$

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

$y = 3 x^{2} + x - 206$

Subtract $9$ from both sides of the equation.

$x = - 9$

$y = 3 x^{2} + x - 206$

$x = - 9$

$y = 3 x^{2} + x - 206$

The final solution is all the values that make $3 (x - 8) (x + 9) = 0$ true.

$x = 8, - 9$

$y = 3 x^{2} + x - 206$

$x = 8, - 9$

$y = 3 x^{2} + x - 206$

Step 3

Replace all occurrences of $x$ with $8$ in each equation.

Replace all occurrences of $x$ in $y = 3 x^{2} + x - 206$ with $8$ .

$y = 3 {(8)}^{2} + 8 - 206$

$x = 8$

Simplify $y = 3 {(8)}^{2} + 8 - 206$ .

Simplify the left side.

Remove parentheses.

$y = 3 {(8)}^{2} + 8 - 206$

$x = 8$

$y = 3 {(8)}^{2} + 8 - 206$

$x = 8$

Simplify the right side.

Simplify $3 {(8)}^{2} + 8 - 206$ .

Simplify each term.

Raise $8$ to the power of $2$ .

$y = 3 \cdot 64 + 8 - 206$

$x = 8$

Multiply $3$ by $64$ .

$y = 192 + 8 - 206$

$x = 8$

$y = 192 + 8 - 206$

$x = 8$

Simplify by adding and subtracting.

Add $192$ and $8$ .

$y = 200 - 206$

$x = 8$

Subtract $206$ from $200$ .

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

Step 4

Replace all occurrences of $x$ with $- 9$ in each equation.

Replace all occurrences of $x$ in $y = 3 x^{2} + x - 206$ with $- 9$ .

$y = 3 {(- 9)}^{2} - 9 - 206$

$x = - 9$

Simplify $y = 3 {(- 9)}^{2} - 9 - 206$ .

Simplify the left side.

Remove parentheses.

$y = 3 {(- 9)}^{2} - 9 - 206$

$x = - 9$

$y = 3 {(- 9)}^{2} - 9 - 206$

$x = - 9$

Simplify the right side.

Simplify $3 {(- 9)}^{2} - 9 - 206$ .

Simplify each term.

Raise $- 9$ to the power of $2$ .

$y = 3 \cdot 81 - 9 - 206$

$x = - 9$

Multiply $3$ by $81$ .

$y = 243 - 9 - 206$

$x = - 9$

$y = 243 - 9 - 206$

$x = - 9$

Simplify by subtracting numbers.

Subtract $9$ from $243$ .

$y = 234 - 206$

$x = - 9$

Subtract $206$ from $234$ .

$y = 28$

$x = - 9$

$y = 28$

$x = - 9$

$y = 28$

$x = - 9$

$y = 28$

$x = - 9$

$y = 28$

$x = - 9$

$y = 28$

$x = - 9$

Step 5

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

$(8, - 6)$

$(- 9, 28)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(8, - 6), (- 9, 28)$

Equation Form:

$x = 8, y = - 6$

$x = - 9, y = 28$

Step 7

Solve the System of Equations y=3x^2+x-206 2x+y=10

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