Calculus Examples

$\sqrt{x^{2} + 10 x + 27}$

Step 1

Set $\sqrt{x^{2} + 10 x + 27}$ equal to $0$ .

$\sqrt{x^{2} + 10 x + 27} = 0$

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + 10 x + 27}}^{2} = 0^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 10 x + 27}$ as ${(x^{2} + 10 x + 27)}^{\frac{1}{2}}$ .

${({(x^{2} + 10 x + 27)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(x^{2} + 10 x + 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(x^{2} + 10 x + 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 10 x + 27)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 10 x + 27)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 10 x + 27)}^{1} = 0^{2}$

Step 2.2.2.1.2

Simplify.

$x^{2} + 10 x + 27 = 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{2} + 10 x + 27 = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2

Substitute the values $a = 1$ , $b = 10$ , and $c = 27$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 27)}}{2 \cdot 1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 27}}{2 \cdot 1}$

Step 2.3.3.1.2

Multiply $- 4 \cdot 1 \cdot 27$ .

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 27}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $27$ .

$x = \frac{- 10 \pm \sqrt{100 - 108}}{2 \cdot 1}$

Step 2.3.3.1.3

Subtract $108$ from $100$ .

$x = \frac{- 10 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 10 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.3.3.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 10 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.3.3.1.6

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

$x = \frac{- 10 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.3.3.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.3.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 10 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.3.3.1.7.2

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

$x = \frac{- 10 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.3.1.8

Pull terms out from under the radical.

$x = \frac{- 10 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 2.3.3.1.9

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

$x = \frac{- 10 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 2.3.3.2

Multiply $2$ by $1$ .

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

Step 2.3.3.3

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

$x = - 5 \pm i \sqrt{2}$

Step 2.3.4

The final answer is the combination of both solutions.

$x = - 5 + i \sqrt{2}, - 5 - i \sqrt{2}$

$x = - 5 \pm i \sqrt{2}$

Step 3