Trigonometry Examples

$2 x^{2} + 12 x - 10 = 0$

Step 1

Add $10$ to both sides of the equation.

$2 x^{2} + 12 x = 10$

Step 2

Divide each term in $2 x^{2} + 12 x = 10$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x^{2} + 12 x = 10$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{12 x}{2} = \frac{10}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{12 x}{2} = \frac{10}{2}$

Step 2.2.1.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{12 x}{2} = \frac{10}{2}$

Step 2.2.1.2

Cancel the common factor of $12$ and $2$ .

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Step 2.2.1.2.1

Factor $2$ out of $12 x$ .

$x^{2} + \frac{2 (6 x)}{2} = \frac{10}{2}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (6 x)}{2 (1)} = \frac{10}{2}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{2 (6 x)}{2 \cdot 1} = \frac{10}{2}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{6 x}{1} = \frac{10}{2}$

Step 2.2.1.2.2.4

Divide $6 x$ by $1$ .

$x^{2} + 6 x = \frac{10}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $10$ by $2$ .

$x^{2} + 6 x = 5$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(3)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + 6 x + {(3)}^{2} = 5 + {(3)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Raise $3$ to the power of $2$ .

$x^{2} + 6 x + 9 = 5 + {(3)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $5 + {(3)}^{2}$ .

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Step 5.2.1.1

Raise $3$ to the power of $2$ .

$x^{2} + 6 x + 9 = 5 + 9$

Step 5.2.1.2

Add $5$ and $9$ .

$x^{2} + 6 x + 9 = 14$

Step 6

Factor the perfect trinomial square into ${(x + 3)}^{2}$ .

${(x + 3)}^{2} = 14$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 3 = \pm \sqrt{14}$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = \pm \sqrt{14} - 3$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{14} - 3$

Decimal Form:

$x = 0.74165738 \dots, - 6.74165738 \dots$