Solve the System of Equations y=-x^2-14x-60 6=x-y

$y = - x^{2} - 14 x - 60$ $6 = x - y$

Replace all occurrences of $y$ with $- x^{2} - 14 x - 60$ in each equation.

Replace all occurrences of $y$ in $6 = x - y$ with $- x^{2} - 14 x - 60$ .

$6 = x - (- x^{2} - 14 x - 60)$

$y = - x^{2} - 14 x - 60$

Simplify the right side.

Simplify $x - (- x^{2} - 14 x - 60)$ .

Simplify each term.

Apply the distributive property.

$6 = x + x^{2} - (- 14 x) + 60$

$y = - x^{2} - 14 x - 60$

Simplify.

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

$6 = x + 1 x^{2} - (- 14 x) + 60$

$y = - x^{2} - 14 x - 60$

Multiply $x^{2}$ by $1$ .

$6 = x + x^{2} - (- 14 x) + 60$

$y = - x^{2} - 14 x - 60$

$6 = x + x^{2} - (- 14 x) + 60$

$y = - x^{2} - 14 x - 60$

Multiply $- 14$ by $- 1$ .

$6 = x + x^{2} + 14 x + 60$

$y = - x^{2} - 14 x - 60$

Multiply $- 1$ by $- 60$ .

$6 = x + x^{2} + 14 x + 60$

$y = - x^{2} - 14 x - 60$

$6 = x + x^{2} + 14 x + 60$

$y = - x^{2} - 14 x - 60$

$6 = x + x^{2} + 14 x + 60$

$y = - x^{2} - 14 x - 60$

Add $x$ and $14 x$ .

$6 = x^{2} + 15 x + 60$

$y = - x^{2} - 14 x - 60$

$6 = x^{2} + 15 x + 60$

$y = - x^{2} - 14 x - 60$

$6 = x^{2} + 15 x + 60$

$y = - x^{2} - 14 x - 60$

$6 = x^{2} + 15 x + 60$

$y = - x^{2} - 14 x - 60$

Solve for $x$ in $6 = x^{2} + 15 x + 60$ .

Rewrite the equation as $x^{2} + 15 x + 60 = 6$ .

$x^{2} + 15 x + 60 = 6$

$y = - x^{2} - 14 x - 60$

Subtract $6$ from both sides of the equation.

$x^{2} + 15 x + 60 - 6 = 0$

$y = - x^{2} - 14 x - 60$

Subtract $6$ from $60$ .

$x^{2} + 15 x + 54 = 0$

$y = - x^{2} - 14 x - 60$

Factor $x^{2} + 15 x + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

$y = - x^{2} - 14 x - 60$

Write the factored form using these integers.

$(x + 6) (x + 9) = 0$

$y = - x^{2} - 14 x - 60$

$(x + 6) (x + 9) = 0$

$y = - x^{2} - 14 x - 60$

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

$x + 6 = 0$

$x + 9 = 0$

$y = - x^{2} - 14 x - 60$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

$y = - x^{2} - 14 x - 60$

Subtract $6$ from both sides of the equation.

$x = - 6$

$y = - x^{2} - 14 x - 60$

$x = - 6$

$y = - x^{2} - 14 x - 60$

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

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

$x + 9 = 0$

$y = - x^{2} - 14 x - 60$

Subtract $9$ from both sides of the equation.

$x = - 9$

$y = - x^{2} - 14 x - 60$

$x = - 9$

$y = - x^{2} - 14 x - 60$

The final solution is all the values that make $(x + 6) (x + 9) = 0$ true.

$x = - 6, - 9$

$y = - x^{2} - 14 x - 60$

$x = - 6, - 9$

$y = - x^{2} - 14 x - 60$

Replace all occurrences of $x$ with $- 6$ in each equation.

Replace all occurrences of $x$ in $y = - x^{2} - 14 x - 60$ with $- 6$ .

$y = - {(- 6)}^{2} - 14 \cdot - 6 - 60$

$x = - 6$

Simplify the right side.

Simplify $- {(- 6)}^{2} - 14 \cdot - 6 - 60$ .

Simplify each term.

Raise $- 6$ to the power of $2$ .

$y = - 1 \cdot 36 - 14 \cdot - 6 - 60$

$x = - 6$

Multiply $- 1$ by $36$ .

$y = - 36 - 14 \cdot - 6 - 60$

$x = - 6$

Multiply $- 14$ by $- 6$ .

$y = - 36 + 84 - 60$

$x = - 6$

$y = - 36 + 84 - 60$

$x = - 6$

Simplify by adding and subtracting.

Add $- 36$ and $84$ .

$y = 48 - 60$

$x = - 6$

Subtract $60$ from $48$ .

$y = - 12$

$x = - 6$

$y = - 12$

$x = - 6$

$y = - 12$

$x = - 6$

$y = - 12$

$x = - 6$

$y = - 12$

$x = - 6$

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

Replace all occurrences of $x$ in $y = - x^{2} - 14 x - 60$ with $- 9$ .

$y = - {(- 9)}^{2} - 14 \cdot - 9 - 60$

$x = - 9$

Simplify the right side.

Simplify $- {(- 9)}^{2} - 14 \cdot - 9 - 60$ .

Simplify each term.

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

$y = - 1 \cdot 81 - 14 \cdot - 9 - 60$

$x = - 9$

Multiply $- 1$ by $81$ .

$y = - 81 - 14 \cdot - 9 - 60$

$x = - 9$

Multiply $- 14$ by $- 9$ .

$y = - 81 + 126 - 60$

$x = - 9$

$y = - 81 + 126 - 60$

$x = - 9$

Simplify by adding and subtracting.

Add $- 81$ and $126$ .

$y = 45 - 60$

$x = - 9$

Subtract $60$ from $45$ .

$y = - 15$

$x = - 9$

$y = - 15$

$x = - 9$

$y = - 15$

$x = - 9$

$y = - 15$

$x = - 9$

$y = - 15$

$x = - 9$

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

$(- 6, - 12)$

$(- 9, - 15)$

The result can be shown in multiple forms.

Point Form:

$(- 6, - 12), (- 9, - 15)$

Equation Form:

$x = - 6, y = - 12$

$x = - 9, y = - 15$

Solve the System of Equations y=-x^2-14x-60 6=x-y

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