Solve the System of Equations y=5x^2-7x-3 y=3x^2+3x+45

$y = 5 x^{2} - 7 x - 3$ $y = 3 x^{2} + 3 x + 45$

Eliminate the equal sides of each equation and combine.

$5 x^{2} - 7 x - 3 = 3 x^{2} + 3 x + 45$

Solve $5 x^{2} - 7 x - 3 = 3 x^{2} + 3 x + 45$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x^{2}$ from both sides of the equation.

$5 x^{2} - 7 x - 3 - 3 x^{2} = 3 x + 45$

Subtract $3 x$ from both sides of the equation.

$5 x^{2} - 7 x - 3 - 3 x^{2} - 3 x = 45$

Subtract $3 x^{2}$ from $5 x^{2}$ .

$2 x^{2} - 7 x - 3 - 3 x = 45$

Subtract $3 x$ from $- 7 x$ .

$2 x^{2} - 10 x - 3 = 45$

Subtract $45$ from both sides of the equation.

$2 x^{2} - 10 x - 3 - 45 = 0$

Subtract $45$ from $- 3$ .

$2 x^{2} - 10 x - 48 = 0$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} - 10 x - 48$ .

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

$2 (x^{2}) - 10 x - 48 = 0$

Factor $2$ out of $- 10 x$ .

$2 (x^{2}) + 2 (- 5 x) - 48 = 0$

Factor $2$ out of $- 48$ .

$2 x^{2} + 2 (- 5 x) + 2 \cdot - 24 = 0$

Factor $2$ out of $2 x^{2} + 2 (- 5 x)$ .

$2 (x^{2} - 5 x) + 2 \cdot - 24 = 0$

Factor $2$ out of $2 (x^{2} - 5 x) + 2 \cdot - 24$ .

$2 (x^{2} - 5 x - 24) = 0$

Factor.

Factor $x^{2} - 5 x - 24$ 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 $- 24$ and whose sum is $- 5$ .

$- 8, 3 + y = 3 x^{2} + 3 x + 45$

Write the factored form using these integers.

$2 ((x - 8) (x + 3)) = 0$

Remove unnecessary parentheses.

$2 (x - 8) (x + 3) = 0$

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

$x - 8 = 0$

$x + 3 = 0 + y = 3 x^{2} + 3 x + 45$

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

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

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $2 (x - 8) (x + 3) = 0$ true.

$x = 8, - 3$

Evaluate $y$ when $x = 8$ .

Substitute $8$ for $x$ .

$y = 3 {(8)}^{2} + 3 (8) + 45$

Substitute $8$ for $x$ in $y = 3 {(8)}^{2} + 3 (8) + 45$ and solve for $y$ .

Remove parentheses.

$y = 3 \cdot 8^{2} + 3 (8) + 45$

Simplify $3 \cdot 8^{2} + 3 (8) + 45$ .

Simplify each term.

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

$y = 3 \cdot 64 + 3 (8) + 45$

Multiply $3$ by $64$ .

$y = 192 + 3 (8) + 45$

Multiply $3$ by $8$ .

$y = 192 + 24 + 45$

Simplify by adding numbers.

Add $192$ and $24$ .

$y = 216 + 45$

Add $216$ and $45$ .

$y = 261$

Evaluate $y$ when $x = - 3$ .

Substitute $- 3$ for $x$ .

$y = 3 {(- 3)}^{2} + 3 (- 3) + 45$

Simplify $3 {(- 3)}^{2} + 3 (- 3) + 45$ .

Simplify each term.

Raise $- 3$ to the power of $2$ .

$y = 3 \cdot 9 + 3 (- 3) + 45$

Multiply $3$ by $9$ .

$y = 27 + 3 (- 3) + 45$

Multiply $3$ by $- 3$ .

$y = 27 - 9 + 45$

Simplify by adding and subtracting.

Subtract $9$ from $27$ .

$y = 18 + 45$

Add $18$ and $45$ .

$y = 63$

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

$(8, 261)$

$(- 3, 63)$

The result can be shown in multiple forms.

Point Form:

$(8, 261), (- 3, 63)$

Equation Form:

$x = 8, y = 261$

$x = - 3, y = 63$

Solve the System of Equations y=5x^2-7x-3 y=3x^2+3x+45

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