Calculus Examples

$x + 2 = \sqrt{2 x - 9} + 4$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{2 x - 9} + 4 = x + 2$

Step 2

Move all terms not containing $\sqrt{2 x - 9}$ to the right side of the equation.

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Step 2.1

Subtract $4$ from both sides of the equation.

$\sqrt{2 x - 9} = x + 2 - 4$

Step 2.2

Subtract $4$ from $2$ .

$\sqrt{2 x - 9} = x - 2$

Step 3

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

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

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 $\sqrt{2 x - 9}$ as ${(2 x - 9)}^{\frac{1}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$2 x - 9 = (x - 2) (x - 2)$

Step 4.3.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$2 x - 9 = x (x - 2) - 2 (x - 2)$

Step 4.3.1.2.2

Apply the distributive property.

$2 x - 9 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 4.3.1.2.3

Apply the distributive property.

$2 x - 9 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $x$ by $x$ .

$2 x - 9 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.3.1.3.1.2

Move $- 2$ to the left of $x$ .

$2 x - 9 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 4.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$2 x - 9 = x^{2} - 2 x - 2 x + 4$

Step 4.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$2 x - 9 = x^{2} - 4 x + 4$

Step 5

Solve for $x$ .

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Step 5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 4 x + 4 = 2 x - 9$

Step 5.2

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

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Step 5.2.1

Subtract $2 x$ from both sides of the equation.

$x^{2} - 4 x + 4 - 2 x = - 9$

Step 5.2.2

Subtract $2 x$ from $- 4 x$ .

$x^{2} - 6 x + 4 = - 9$

Step 5.3

Move all terms to the left side of the equation and simplify.

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Step 5.3.1

Add $9$ to both sides of the equation.

$x^{2} - 6 x + 4 + 9 = 0$

Step 5.3.2

Add $4$ and $9$ .

$x^{2} - 6 x + 13 = 0$

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

Substitute the values $a = 1$ , $b = - 6$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 5.6.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Step 5.6.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Step 5.6.1.3

Subtract $52$ from $36$ .

$x = \frac{6 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 5.6.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 5.6.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 5.6.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 5.6.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 5.6.1.8

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

$x = \frac{6 \pm i \cdot 4}{2 \cdot 1}$

Step 5.6.1.9

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i}{2 \cdot 1}$

Step 5.6.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 i}{2}$

Step 5.6.3

Simplify $\frac{6 \pm 4 i}{2}$ .

$x = 3 \pm 2 i$

Step 5.7

The final answer is the combination of both solutions.

$x = 3 + 2 i, 3 - 2 i$