Trigonometry Examples

$x^{2} + 10 x - 107 = - 7$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $107$ to both sides of the equation.

$x^{2} + 10 x = - 7 + 107$

Step 1.2

Add $- 7$ and $107$ .

$x^{2} + 10 x = 100$

Step 2

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

${(\frac{b}{2})}^{2} = {(5)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 10 x + {(5)}^{2} = 100 + {(5)}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Raise $5$ to the power of $2$ .

$x^{2} + 10 x + 25 = 100 + {(5)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $100 + {(5)}^{2}$ .

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Step 4.2.1.1

Raise $5$ to the power of $2$ .

$x^{2} + 10 x + 25 = 100 + 25$

Step 4.2.1.2

Add $100$ and $25$ .

$x^{2} + 10 x + 25 = 125$

Step 5

Factor the perfect trinomial square into ${(x + 5)}^{2}$ .

${(x + 5)}^{2} = 125$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x + 5 = \pm \sqrt{125}$

Step 6.2

Simplify $\pm \sqrt{125}$ .

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Step 6.2.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 6.2.1.1

Factor $25$ out of $125$ .

$x + 5 = \pm \sqrt{25 (5)}$

Step 6.2.1.2

Rewrite $25$ as $5^{2}$ .

$x + 5 = \pm \sqrt{5^{2} \cdot 5}$

Step 6.2.2

Pull terms out from under the radical.

$x + 5 = \pm 5 \sqrt{5}$

Step 6.3

Subtract $5$ from both sides of the equation.

$x = \pm 5 \sqrt{5} - 5$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 5 \sqrt{5} - 5$

Decimal Form:

$x = 6.18033988 \dots, - 16.18033988 \dots$