Calculus Examples

$\frac{1.1 x - \sqrt{x^{2} + 9}}{\sqrt{x^{2} + 9}} = 0$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$0 \cdot (\sqrt{x^{2} + 9}) = 1.1 x - \sqrt{x^{2} + 9}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $0 \cdot (\sqrt{x^{2} + 9})$ .

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Step 1.2.1.1

Remove parentheses.

$0 \cdot (\sqrt{x^{2} + 9}) = 1.1 x - \sqrt{x^{2} + 9}$

Step 1.2.1.2

Multiply $0$ by $\sqrt{x^{2} + 9}$ .

$0 = 1.1 x - \sqrt{x^{2} + 9}$

Step 2

Solve for $- \sqrt{x^{2} + 9}$ .

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Step 2.1

Rewrite the equation as $1.1 x - \sqrt{x^{2} + 9} = 0$ .

$1.1 x - \sqrt{x^{2} + 9} = 0$

Step 2.2

Subtract $1.1 x$ from both sides of the equation.

$- \sqrt{x^{2} + 9} = - 1.1 x$

Step 3

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

${(- \sqrt{x^{2} + 9})}^{2} = {(- 1.1 x)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 9}$ as ${(x^{2} + 9)}^{\frac{1}{2}}$ .

${(- {(x^{2} + 9)}^{\frac{1}{2}})}^{2} = {(- 1.1 x)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(x^{2} + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $- {(x^{2} + 9)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x^{2} + 9)}^{\frac{1}{2}})}^{2} = {(- 1.1 x)}^{2}$

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(x^{2} + 9)}^{\frac{1}{2}})}^{2} = {(- 1.1 x)}^{2}$

Step 4.2.1.3

Multiply ${({(x^{2} + 9)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x^{2} + 9)}^{\frac{1}{2}})}^{2} = {(- 1.1 x)}^{2}$

Step 4.2.1.4

Multiply the exponents in ${({(x^{2} + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

${(x^{2} + 9)}^{\frac{1}{2} \cdot 2} = {(- 1.1 x)}^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

${(x^{2} + 9)}^{\frac{1}{2} \cdot 2} = {(- 1.1 x)}^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

${(x^{2} + 9)}^{1} = {(- 1.1 x)}^{2}$

Step 4.2.1.5

Simplify.

$x^{2} + 9 = {(- 1.1 x)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- 1.1 x)}^{2}$ .

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Step 4.3.1.1

Apply the product rule to $- 1.1 x$ .

$x^{2} + 9 = {(- 1.1)}^{2} x^{2}$

Step 4.3.1.2

Raise $- 1.1$ to the power of $2$ .

$x^{2} + 9 = 1.21 x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Move all terms containing $x$ to the left side of the equation.

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Step 5.1.1

Subtract $1.21 x^{2}$ from both sides of the equation.

$x^{2} + 9 - 1.21 x^{2} = 0$

Step 5.1.2

Subtract $1.21 x^{2}$ from $x^{2}$ .

$- 0.21 x^{2} + 9 = 0$

Step 5.2

Subtract $9$ from both sides of the equation.

$- 0.21 x^{2} = - 9$

Step 5.3

Divide each term in $- 0.21 x^{2} = - 9$ by $- 0.21$ and simplify.

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Step 5.3.1

Divide each term in $- 0.21 x^{2} = - 9$ by $- 0.21$ .

$\frac{- 0.21 x^{2}}{- 0.21} = \frac{- 9}{- 0.21}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 0.21$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 0.21 x^{2}}{- 0.21} = \frac{- 9}{- 0.21}$

Step 5.3.2.1.2

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

$x^{2} = \frac{- 9}{- 0.21}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 9$ by $- 0.21$ .

$x^{2} = 42.\overline{857142}$

Step 5.4

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

$x = \pm \sqrt{42.\overline{857142}}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{42.\overline{857142}}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{42.\overline{857142}}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{42.\overline{857142}}, - \sqrt{42.\overline{857142}}$

Step 6

Exclude the solutions that do not make $\frac{1.1 x - \sqrt{x^{2} + 9}}{\sqrt{x^{2} + 9}} = 0$ true.

$x = \sqrt{42.\overline{857142}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{42.\overline{857142}}$

Decimal Form:

$x = 6.54653670 \dots$