Algebra Examples

3 x^{2} + 5 x + 2 \frac{1}{12}

$3 x^{2} + 5 x + 2 \frac{1}{12}$

Step 1

Set

3 x^{2} + 5 x + 2 \frac{1}{12}

$3 x^{2} + 5 x + 2 \frac{1}{12}$ equal to

0

$0$ .

3 x^{2} + 5 x + 2 \frac{1}{12} = 0

$3 x^{2} + 5 x + 2 \frac{1}{12} = 0$

Step 2

Simplify the left side.

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Step 2.1

Convert

2 \frac{1}{12}

$2 \frac{1}{12}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

3 x^{2} + 5 x + 2 + \frac{1}{12} = 0

$3 x^{2} + 5 x + 2 + \frac{1}{12} = 0$

Step 2.1.2

Add

2

$2$ and

\frac{1}{12}

$\frac{1}{12}$ .

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Step 2.1.2.1

To write

2

$2$ as a fraction with a common denominator, multiply by

\frac{12}{12}

$\frac{12}{12}$ .

3 x^{2} + 5 x + 2 \cdot \frac{12}{12} + \frac{1}{12} = 0

$3 x^{2} + 5 x + 2 \cdot \frac{12}{12} + \frac{1}{12} = 0$

Step 2.1.2.2

Combine

2

$2$ and

\frac{12}{12}

$\frac{12}{12}$ .

3 x^{2} + 5 x + \frac{2 \cdot 12}{12} + \frac{1}{12} = 0

$3 x^{2} + 5 x + \frac{2 \cdot 12}{12} + \frac{1}{12} = 0$

Step 2.1.2.3

Combine the numerators over the common denominator.

3 x^{2} + 5 x + \frac{2 \cdot 12 + 1}{12} = 0

$3 x^{2} + 5 x + \frac{2 \cdot 12 + 1}{12} = 0$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply

2

$2$ by

12

$12$ .

3 x^{2} + 5 x + \frac{24 + 1}{12} = 0

$3 x^{2} + 5 x + \frac{24 + 1}{12} = 0$

Step 2.1.2.4.2

Add

24

$24$ and

1

$1$ .

3 x^{2} + 5 x + \frac{25}{12} = 0

$3 x^{2} + 5 x + \frac{25}{12} = 0$

3 x^{2} + 5 x + \frac{25}{12} = 0

$3 x^{2} + 5 x + \frac{25}{12} = 0$

3 x^{2} + 5 x + \frac{25}{12} = 0

$3 x^{2} + 5 x + \frac{25}{12} = 0$

3 x^{2} + 5 x + \frac{25}{12} = 0

$3 x^{2} + 5 x + \frac{25}{12} = 0$

3 x^{2} + 5 x + \frac{25}{12} = 0

$3 x^{2} + 5 x + \frac{25}{12} = 0$

Step 3

Multiply through by the least common denominator

12

$12$ , then simplify.

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Step 3.1

Apply the distributive property.

12 (3 x^{2}) + 12 (5 x) + 12 (\frac{25}{12}) = 0

$12 (3 x^{2}) + 12 (5 x) + 12 (\frac{25}{12}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply

3

$3$ by

12

$12$ .

36 x^{2} + 12 (5 x) + 12 (\frac{25}{12}) = 0

$36 x^{2} + 12 (5 x) + 12 (\frac{25}{12}) = 0$

Step 3.2.2

Multiply

5

$5$ by

12

$12$ .

36 x^{2} + 60 x + 12 (\frac{25}{12}) = 0

$36 x^{2} + 60 x + 12 (\frac{25}{12}) = 0$

Step 3.2.3

Cancel the common factor of

12

$12$ .

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Step 3.2.3.1

Cancel the common factor.

$36 x^{2} + 60 x + 12 (\frac{25}{12}) = 0$

Step 3.2.3.2

Rewrite the expression.

$36 x^{2} + 60 x + 25 = 0$

$36 x^{2} + 60 x + 25 = 0$

$36 x^{2} + 60 x + 25 = 0$

$36 x^{2} + 60 x + 25 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values

$a = 36$ ,

$b = 60$ , and

$c = 25$ into the quadratic formula and solve for

$x$ .

$\frac{- 60 \pm \sqrt{60^{2} - 4 \cdot (36 \cdot 25)}}{2 \cdot 36}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise

$60$ to the power of

$2$ .

$x = \frac{- 60 \pm \sqrt{3600 - 4 \cdot 36 \cdot 25}}{2 \cdot 36}$

Step 6.1.2

Multiply

$- 4 \cdot 36 \cdot 25$ .

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Step 6.1.2.1

Multiply

$- 4$ by

$36$ .

$x = \frac{- 60 \pm \sqrt{3600 - 144 \cdot 25}}{2 \cdot 36}$

Step 6.1.2.2

Multiply

$- 144$ by

$25$ .

$x = \frac{- 60 \pm \sqrt{3600 - 3600}}{2 \cdot 36}$

$x = \frac{- 60 \pm \sqrt{3600 - 3600}}{2 \cdot 36}$

Step 6.1.3

Subtract

$3600$ from

$3600$ .

$x = \frac{- 60 \pm \sqrt{0}}{2 \cdot 36}$

Step 6.1.4

Rewrite

$0$ as

$0^{2}$ .

$x = \frac{- 60 \pm \sqrt{0^{2}}}{2 \cdot 36}$

Step 6.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 60 \pm 0}{2 \cdot 36}$

Step 6.1.6

$- 60$ plus or minus

$0$ is

$- 60$ .

$x = \frac{- 60}{2 \cdot 36}$

$x = \frac{- 60}{2 \cdot 36}$

Step 6.2

Multiply

$2$ by

$36$ .

$x = \frac{- 60}{72}$

Step 6.3

Cancel the common factor of

$- 60$ and

$72$ .

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Step 6.3.1

Factor

$12$ out of

$- 60$ .

$x = \frac{12 (- 5)}{72}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor

$12$ out of

$72$ .

$x = \frac{12 \cdot - 5}{12 \cdot 6}$

Step 6.3.2.2

Cancel the common factor.

$x = \frac{12 \cdot - 5}{12 \cdot 6}$

Step 6.3.2.3

Rewrite the expression.

$x = \frac{- 5}{6}$

$x = \frac{- 5}{6}$

$x = \frac{- 5}{6}$

Step 6.4

Move the negative in front of the fraction.

$x = - \frac{5}{6}$

$x = - \frac{5}{6}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{5}{6}$ Double roots

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$