Precalculus Examples

$\frac{2}{5} x^{2} + 2 x + \frac{11}{5} = 0$

Step 1

Simplify the left side.

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Step 1.1

Combine $\frac{2}{5}$ and $x^{2}$ .

$\frac{2 x^{2}}{5} + 2 x + \frac{11}{5} = 0$

Step 2

Multiply through by the least common denominator $5$ , then simplify.

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Step 2.1

Apply the distributive property.

$5 (\frac{2 x^{2}}{5}) + 5 (2 x) + 5 (\frac{11}{5}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$5 (\frac{2 x^{2}}{5}) + 5 (2 x) + 5 (\frac{11}{5}) = 0$

Step 2.2.1.2

Rewrite the expression.

$2 x^{2} + 5 (2 x) + 5 (\frac{11}{5}) = 0$

Step 2.2.2

Multiply $2$ by $5$ .

$2 x^{2} + 10 x + 5 (\frac{11}{5}) = 0$

Step 2.2.3

Cancel the common factor of $5$ .

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Step 2.2.3.1

Cancel the common factor.

$2 x^{2} + 10 x + 5 (\frac{11}{5}) = 0$

Step 2.2.3.2

Rewrite the expression.

$2 x^{2} + 10 x + 11 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = 10$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot 11)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 11}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 8 \cdot 11}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $11$ .

$x = \frac{- 10 \pm \sqrt{100 - 88}}{2 \cdot 2}$

Step 5.1.3

Subtract $88$ from $100$ .

$x = \frac{- 10 \pm \sqrt{12}}{2 \cdot 2}$

Step 5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 10 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 10 \pm 2 \sqrt{3}}{4}$

Step 5.3

Simplify $\frac{- 10 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{- 5 \pm \sqrt{3}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm \sqrt{3}}{2}$

Decimal Form:

$x = - 1.63397459 \dots, - 3.36602540 \dots$