Calculus Examples

$x - \sqrt{9 - 3 x} = 0$

Step 1

Subtract $x$ from both sides of the equation.

$- \sqrt{9 - 3 x} = - x$

Step 2

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

${(- \sqrt{9 - 3 x})}^{2} = {(- x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - 3 x}$ as ${(9 - 3 x)}^{\frac{1}{2}}$ .

${(- {(9 - 3 x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(9 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- {(9 - 3 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(9 - 3 x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 3.2.1.2

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

$1 {({(9 - 3 x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 3.2.1.3

Multiply ${({(9 - 3 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(9 - 3 x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 3.2.1.4

Multiply the exponents in ${({(9 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

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

${(9 - 3 x)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

${(9 - 3 x)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

${(9 - 3 x)}^{1} = {(- x)}^{2}$

Step 3.2.1.5

Simplify.

$9 - 3 x = {(- x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $- x$ .

$9 - 3 x = {(- 1)}^{2} x^{2}$

Step 3.3.1.2

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

$9 - 3 x = 1 x^{2}$

Step 3.3.1.3

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

$9 - 3 x = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$9 - 3 x - x^{2} = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = - 1$ , $b = - 3$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 1 \cdot 9)}}{2 \cdot - 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Step 4.4.1.2

Multiply $- 4 \cdot - 1 \cdot 9$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot 9}}{2 \cdot - 1}$

Step 4.4.1.2.2

Multiply $4$ by $9$ .

$x = \frac{3 \pm \sqrt{9 + 36}}{2 \cdot - 1}$

Step 4.4.1.3

Add $9$ and $36$ .

$x = \frac{3 \pm \sqrt{45}}{2 \cdot - 1}$

Step 4.4.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.4.1.4.1

Factor $9$ out of $45$ .

$x = \frac{3 \pm \sqrt{9 (5)}}{2 \cdot - 1}$

Step 4.4.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{3 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot - 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{3 \pm 3 \sqrt{5}}{2 \cdot - 1}$

Step 4.4.2

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm 3 \sqrt{5}}{- 2}$

Step 4.4.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm 3 \sqrt{5}}{2}$

Step 4.5

The final answer is the combination of both solutions.

$x = - \frac{3 + 3 \sqrt{5}}{2}, - \frac{3 - 3 \sqrt{5}}{2}$

Step 5

Exclude the solutions that do not make $x - \sqrt{9 - 3 x} = 0$ true.

$x = - \frac{3 - 3 \sqrt{5}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - 3 \sqrt{5}}{2}$

Decimal Form:

$x = 1.85410196 \dots$