Trigonometry Examples

$\frac{3}{4} x^{2} - 2 x + \frac{1}{2} = 0$

Step 1

Combine $\frac{3}{4}$ and $x^{2}$ .

$\frac{3 x^{2}}{4} - 2 x + \frac{1}{2} = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

$4 (\frac{3 x^{2}}{4}) + 4 (- 2 x) + 4 (\frac{1}{2}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$4 (\frac{3 x^{2}}{4}) + 4 (- 2 x) + 4 (\frac{1}{2}) = 0$

Step 2.2.1.2

Rewrite the expression.

$3 x^{2} + 4 (- 2 x) + 4 (\frac{1}{2}) = 0$

Step 2.2.2

Multiply $- 2$ by $4$ .

$3 x^{2} - 8 x + 4 (\frac{1}{2}) = 0$

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Factor $2$ out of $4$ .

$3 x^{2} - 8 x + 2 (2) (\frac{1}{2}) = 0$

Step 2.2.3.2

Cancel the common factor.

$3 x^{2} - 8 x + 2 \cdot (2 (\frac{1}{2})) = 0$

Step 2.2.3.3

Rewrite the expression.

$3 x^{2} - 8 x + 2 = 0$

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 $a = 3$ , $b = - 8$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot 2)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 2}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $2$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 3}$

Step 5.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 3}$

Step 5.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 5.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 3}$

Step 5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 3}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{10}}{6}$

Step 5.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{6}$ .

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

Step 6

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{10}}{3}, \frac{4 - \sqrt{10}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 + \sqrt{10}}{3}, \frac{4 - \sqrt{10}}{3}$

Decimal Form:

$x = 2.38742588 \dots, 0.27924077 \dots$