Trigonometry Examples

$x^{2} - 12 x + 34 = - 6$

Step 1

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

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Step 1.1

Subtract $34$ from both sides of the equation.

$x^{2} - 12 x = - 6 - 34$

Step 1.2

Subtract $34$ from $- 6$ .

$x^{2} - 12 x = - 40$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 6)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 12 x + {(- 6)}^{2} = - 40 + {(- 6)}^{2}$

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 $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = - 40 + {(- 6)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 40 + {(- 6)}^{2}$ .

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Step 4.2.1.1

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = - 40 + 36$

Step 4.2.1.2

Add $- 40$ and $36$ .

$x^{2} - 12 x + 36 = - 4$

Step 5

Factor the perfect trinomial square into ${(x - 6)}^{2}$ .

${(x - 6)}^{2} = - 4$

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.

$x - 6 = \pm \sqrt{- 4}$

Step 6.2

Simplify $\pm \sqrt{- 4}$ .

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Step 6.2.1

Rewrite $- 4$ as $- 1 (4)$ .

$x - 6 = \pm \sqrt{- 1 (4)}$

Step 6.2.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x - 6 = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 6.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x - 6 = \pm i \cdot \sqrt{4}$

Step 6.2.4

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

$x - 6 = \pm i \cdot \sqrt{2^{2}}$

Step 6.2.5

Pull terms out from under the radical, assuming positive real numbers.

$x - 6 = \pm i \cdot 2$

Step 6.2.6

Move $2$ to the left of $i$ .

$x - 6 = \pm 2 i$

Step 6.3

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

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Step 6.3.1

Add $6$ to both sides of the equation.

$x = \pm 2 i + 6$

Step 6.3.2

Reorder $\pm 2 i$ and $6$ .

$x = 6 \pm 2 i$