Algebra Examples

$y = - x^{2} - 6 x - 9$

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 $x$ .

$0 = - x^{2} - 6 x - 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- x^{2} - 6 x - 9 = 0$ .

$- x^{2} - 6 x - 9 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 1$ out of $- x^{2} - 6 x - 9$ .

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Step 1.2.2.1.1

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

$- (x^{2}) - 6 x - 9 = 0$

Step 1.2.2.1.2

Factor $- 1$ out of $- 6 x$ .

$- (x^{2}) - (6 x) - 9 = 0$

Step 1.2.2.1.3

Rewrite $- 9$ as $- 1 (9)$ .

$- (x^{2}) - (6 x) - 1 \cdot 9 = 0$

Step 1.2.2.1.4

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

$- (x^{2} + 6 x) - 1 \cdot 9 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (x^{2} + 6 x) - 1 (9)$ .

$- (x^{2} + 6 x + 9) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.2.2.3

Rewrite the polynomial.

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

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 1.2.3

Divide each term in $- {(x + 3)}^{2} = 0$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- {(x + 3)}^{2} = 0$ by $- 1$ .

$\frac{- {(x + 3)}^{2}}{- 1} = \frac{0}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x + 3)}^{2}}{1} = \frac{0}{- 1}$

Step 1.2.3.2.2

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{0}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 1$ .

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

Step 1.2.4

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

$x + 3 = 0$

Step 1.2.5

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = - {(0)}^{2} - 6 \cdot 0 - 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 0^{2} - 6 \cdot 0 - 9$

Step 2.2.2

Remove parentheses.

$y = - {(0)}^{2} - 6 \cdot 0 - 9$

Step 2.2.3

Simplify $- {(0)}^{2} - 6 \cdot 0 - 9$ .

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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 $0$ .

$y = - 0 - 6 \cdot 0 - 9$

Step 2.2.3.1.2

Multiply $- 1$ by $0$ .

$y = 0 - 6 \cdot 0 - 9$

Step 2.2.3.1.3

Multiply $- 6$ by $0$ .

$y = 0 + 0 - 9$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 9$

Step 2.2.3.2.2

Subtract $9$ from $0$ .

$y = - 9$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 9)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 9)$

Step 4