Algebra Examples

$y = - x^{2} - 5 x - 6$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - x^{2} - 5 x - 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- x^{2} - 5 x - 6 = 0$ .

$- x^{2} - 5 x - 6 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 1$ out of $- x^{2} - 5 x - 6$ .

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Step 1.2.2.1.1

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

$- (x^{2}) - 5 x - 6 = 0$

Step 1.2.2.1.2

Factor $- 1$ out of $- 5 x$ .

$- (x^{2}) - (5 x) - 6 = 0$

Step 1.2.2.1.3

Rewrite $- 6$ as $- 1 (6)$ .

$- (x^{2}) - (5 x) - 1 \cdot 6 = 0$

Step 1.2.2.1.4

Factor $- 1$ out of $- (x^{2}) - (5 x)$ .

$- (x^{2} + 5 x) - 1 \cdot 6 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (x^{2} + 5 x) - 1 (6)$ .

$- (x^{2} + 5 x + 6) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 1.2.2.2.1.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.2.2.2.1.2

Write the factored form using these integers.

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

Step 1.2.2.2.2

Remove unnecessary parentheses.

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

Step 1.2.3

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

$x + 2 = 0$

$x + 3 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x + 2 = 0$

Step 1.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 3 = 0$

Step 1.2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

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

$x = - 2, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - {(0)}^{2} - 5 \cdot 0 - 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 0^{2} - 5 \cdot 0 - 6$

Step 2.2.2

Remove parentheses.

$y = - {(0)}^{2} - 5 \cdot 0 - 6$

Step 2.2.3

Simplify $- {(0)}^{2} - 5 \cdot 0 - 6$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = - 0 - 5 \cdot 0 - 6$

Step 2.2.3.1.2

Multiply $- 1$ by $0$ .

$y = 0 - 5 \cdot 0 - 6$

Step 2.2.3.1.3

Multiply $- 5$ by $0$ .

$y = 0 + 0 - 6$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 6$

Step 2.2.3.2.2

Subtract $6$ from $0$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0), (- 3, 0)$

y-intercept(s): $(0, - 6)$

Step 4