Algebra Examples

y = x^{2} + 5 x - 9

$y = x^{2} + 5 x - 9$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = x^{2} + 5 x - 9

$0 = x^{2} + 5 x - 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as

x^{2} + 5 x - 9 = 0

$x^{2} + 5 x - 9 = 0$ .

x^{2} + 5 x - 9 = 0

$x^{2} + 5 x - 9 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.3

Substitute the values

a = 1

$a = 1$ ,

b = 5

$b = 5$ , and

c = - 9

$c = - 9$ into the quadratic formula and solve for

x

$x$ .

\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise

5

$5$ to the power of

2

$2$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 1.2.4.1.2

Multiply

- 4 \cdot 1 \cdot - 9

$- 4 \cdot 1 \cdot - 9$ .

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Step 1.2.4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}$

Step 1.2.4.1.2.2

Multiply

- 4

$- 4$ by

- 9

$- 9$ .

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

Step 1.2.4.1.3

Add

25

$25$ and

36

$36$ .

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

Step 1.2.4.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{61}}{2}

$x = \frac{- 5 \pm \sqrt{61}}{2}$

x = \frac{- 5 \pm \sqrt{61}}{2}

$x = \frac{- 5 \pm \sqrt{61}}{2}$

Step 1.2.5

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise

5

$5$ to the power of

2

$2$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply

- 4 \cdot 1 \cdot - 9

$- 4 \cdot 1 \cdot - 9$ .

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Step 1.2.5.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply

- 4

$- 4$ by

- 9

$- 9$ .

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

Step 1.2.5.1.3

Add

25

$25$ and

36

$36$ .

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

Step 1.2.5.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{61}}{2}

$x = \frac{- 5 \pm \sqrt{61}}{2}$

Step 1.2.5.3

Change the

\pm

$\pm$ to

+

$+$ .

x = \frac{- 5 + \sqrt{61}}{2}

$x = \frac{- 5 + \sqrt{61}}{2}$

Step 1.2.5.4

Rewrite

- 5

$- 5$ as

- 1 (5)

$- 1 (5)$ .

x = \frac{- 1 \cdot 5 + \sqrt{61}}{2}

$x = \frac{- 1 \cdot 5 + \sqrt{61}}{2}$

Step 1.2.5.5

Factor

- 1

$- 1$ out of

\sqrt{61}

$\sqrt{61}$ .

x = \frac{- 1 \cdot 5 - 1 (- \sqrt{61})}{2}

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{61})}{2}$

Step 1.2.5.6

Factor

- 1

$- 1$ out of

- 1 (5) - 1 (- \sqrt{61})

$- 1 (5) - 1 (- \sqrt{61})$ .

x = \frac{- 1 (5 - \sqrt{61})}{2}

$x = \frac{- 1 (5 - \sqrt{61})}{2}$

Step 1.2.5.7

Move the negative in front of the fraction.

x = - \frac{5 - \sqrt{61}}{2}

$x = - \frac{5 - \sqrt{61}}{2}$

x = - \frac{5 - \sqrt{61}}{2}

$x = - \frac{5 - \sqrt{61}}{2}$

Step 1.2.6

Simplify the expression to solve for the

-

$-$ portion of the

\pm

$\pm$ .

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise

5

$5$ to the power of

2

$2$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply

- 4 \cdot 1 \cdot - 9

$- 4 \cdot 1 \cdot - 9$ .

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Step 1.2.6.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply

- 4

$- 4$ by

- 9

$- 9$ .

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

Step 1.2.6.1.3

Add

25

$25$ and

36

$36$ .

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

Step 1.2.6.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 5 \pm \sqrt{61}}{2}

$x = \frac{- 5 \pm \sqrt{61}}{2}$

Step 1.2.6.3

Change the

\pm

$\pm$ to

-

$-$ .

x = \frac{- 5 - \sqrt{61}}{2}

$x = \frac{- 5 - \sqrt{61}}{2}$

Step 1.2.6.4

Rewrite

- 5

$- 5$ as

- 1 (5)

$- 1 (5)$ .

x = \frac{- 1 \cdot 5 - \sqrt{61}}{2}

$x = \frac{- 1 \cdot 5 - \sqrt{61}}{2}$

Step 1.2.6.5

Factor

- 1

$- 1$ out of

- \sqrt{61}

$- \sqrt{61}$ .

x = \frac{- 1 \cdot 5 - (\sqrt{61})}{2}

$x = \frac{- 1 \cdot 5 - (\sqrt{61})}{2}$

Step 1.2.6.6

Factor

- 1

$- 1$ out of

- 1 (5) - (\sqrt{61})

$- 1 (5) - (\sqrt{61})$ .

x = \frac{- 1 (5 + \sqrt{61})}{2}

$x = \frac{- 1 (5 + \sqrt{61})}{2}$

Step 1.2.6.7

Move the negative in front of the fraction.

x = - \frac{5 + \sqrt{61}}{2}

$x = - \frac{5 + \sqrt{61}}{2}$

x = - \frac{5 + \sqrt{61}}{2}

$x = - \frac{5 + \sqrt{61}}{2}$

Step 1.2.7

The final answer is the combination of both solutions.

x = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}

$x = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}$

x = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}

$x = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s):

(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)

$(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)$

x-intercept(s):

(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)

$(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y = {(0)}^{2} + 5 (0) - 9

$y = {(0)}^{2} + 5 (0) - 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

y = 0^{2} + 5 (0) - 9

$y = 0^{2} + 5 (0) - 9$

Step 2.2.2

Remove parentheses.

y = {(0)}^{2} + 5 (0) - 9

$y = {(0)}^{2} + 5 (0) - 9$

Step 2.2.3

Simplify

{(0)}^{2} + 5 (0) - 9

${(0)}^{2} + 5 (0) - 9$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

y = 0 + 5 (0) - 9

$y = 0 + 5 (0) - 9$

Step 2.2.3.1.2

Multiply

5

$5$ by

0

$0$ .

y = 0 + 0 - 9

$y = 0 + 0 - 9$

y = 0 + 0 - 9

$y = 0 + 0 - 9$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add

0

$0$ and

0

$0$ .

y = 0 - 9

$y = 0 - 9$

Step 2.2.3.2.2

Subtract

9

$9$ from

0

$0$ .

y = - 9

$y = - 9$

y = - 9

$y = - 9$

y = - 9

$y = - 9$

y = - 9

$y = - 9$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

(0, - 9)

$(0, - 9)$

y-intercept(s):

(0, - 9)

$(0, - 9)$

Step 3

List the intersections.

x-intercept(s):

(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)

$(- \frac{5 - \sqrt{61}}{2}, 0), (- \frac{5 + \sqrt{61}}{2}, 0)$

y-intercept(s):

(0, - 9)

$(0, - 9)$

Step 4