Algebra Examples

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

Step 1

Set $3 x^{2} + 5 x + 2 \frac{1}{12}$ equal to $0$ .

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

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Add $2$ and $\frac{1}{12}$ .

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Step 2.1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

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

Step 2.1.2.2

Combine $2$ and $\frac{12}{12}$ .

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

Step 2.1.2.3

Combine the numerators over the common denominator.

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

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $2$ by $12$ .

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

Step 2.1.2.4.2

Add $24$ and $1$ .

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

Step 3

Multiply through by the least common denominator $12$ , then simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $12$ .

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

Step 3.2.2

Multiply $5$ by $12$ .

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

Step 3.2.3

Cancel the common factor of $12$ .

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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.

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

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 $25$ .

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

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 $- 60$ .

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

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.

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

Step 6.4

Move the negative in front of the fraction.

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

Step 7

The final answer is the combination of both solutions.

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