Algebra Examples

$6 \cdot 2^{x^{2} - 2 x + 2} = 4 \cdot 3^{x^{2} - 2 x + 2}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (6 \cdot 2^{x^{2} - 2 x + 2}) = \ln (4 \cdot 3^{x^{2} - 2 x + 2})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (6 \cdot 2^{x^{2} - 2 x + 2})$ as $\ln (6) + \ln (2^{x^{2} - 2 x + 2})$ .

$\ln (6) + \ln (2^{x^{2} - 2 x + 2}) = \ln (4 \cdot 3^{x^{2} - 2 x + 2})$

Step 2.2

Expand $\ln (2^{x^{2} - 2 x + 2})$ by moving $x^{2} - 2 x + 2$ outside the logarithm.

$\ln (6) + (x^{2} - 2 x + 2) \ln (2) = \ln (4 \cdot 3^{x^{2} - 2 x + 2})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (4 \cdot 3^{x^{2} - 2 x + 2})$ as $\ln (4) + \ln (3^{x^{2} - 2 x + 2})$ .

$\ln (6) + (x^{2} - 2 x + 2) \ln (2) = \ln (4) + \ln (3^{x^{2} - 2 x + 2})$

Step 3.2

Expand $\ln (3^{x^{2} - 2 x + 2})$ by moving $x^{2} - 2 x + 2$ outside the logarithm.

$\ln (6) + (x^{2} - 2 x + 2) \ln (2) = \ln (4) + (x^{2} - 2 x + 2) \ln (3)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$\ln (6) + x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) = \ln (4) + (x^{2} - 2 x + 2) \ln (3)$

Step 5

Simplify the right side.

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Step 5.1

Apply the distributive property.

$\ln (6) + x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) = \ln (4) + x^{2} \ln (3) - 2 x \ln (3) + 2 \ln (3)$

Step 6

Move $\ln (6)$ .

$x^{2} \ln (2) - 2 x \ln (2) + \ln (6) + 2 \ln (2) = \ln (4) + x^{2} \ln (3) - 2 x \ln (3) + 2 \ln (3)$

Step 7

Move $\ln (4)$ .

$x^{2} \ln (2) - 2 x \ln (2) + \ln (6) + 2 \ln (2) = x^{2} \ln (3) - 2 x \ln (3) + \ln (4) + 2 \ln (3)$

Step 8

Move all the terms containing a logarithm to the left side of the equation.

$x^{2} \ln (2) - 2 x \ln (2) + \ln (6) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) - \ln (4) - 2 \ln (3) = 0$

Step 9

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) + \ln (\frac{6}{4}) - 2 \ln (3) = 0$

Step 10

Cancel the common factor of $6$ and $4$ .

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Step 10.1

Factor $2$ out of $6$ .

$x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) + \ln (\frac{2 (3)}{4}) - 2 \ln (3) = 0$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $4$ .

$x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) + \ln (\frac{2 \cdot 3}{2 \cdot 2}) - 2 \ln (3) = 0$

Step 10.2.2

Cancel the common factor.

$x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) + \ln (\frac{2 \cdot 3}{2 \cdot 2}) - 2 \ln (3) = 0$

Step 10.2.3

Rewrite the expression.

$x^{2} \ln (2) - 2 x \ln (2) + 2 \ln (2) - x^{2} \ln (3) + 2 x \ln (3) + \ln (\frac{3}{2}) - 2 \ln (3) = 0$

Step 11

Use the quadratic formula to find the solutions.

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

Step 12

Substitute the values $a = \ln (2) - \ln (3)$ , $b = - 2 \ln (2) + 2 \ln (3)$ , and $c = 2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3)$ into the quadratic formula and solve for $x$ .

$\frac{- (- 2 \ln (2) + 2 \ln (3)) \pm \sqrt{{(- 2 \ln (2) + 2 \ln (3))}^{2} - 4 \cdot ((\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13

Simplify the numerator.

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Step 13.1

Apply the distributive property.

$x = \frac{- (- 2 \ln (2)) - (2 \ln (3)) \pm \sqrt{{(- 2 \ln (2) + 2 \ln (3))}^{2} - 4 \cdot (\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.2

Multiply $- 2$ by $- 1$ .

$x = \frac{2 \ln (2) - (2 \ln (3)) \pm \sqrt{{(- 2 \ln (2) + 2 \ln (3))}^{2} - 4 \cdot (\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.3

Multiply $2$ by $- 1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{{(- 2 \ln (2) + 2 \ln (3))}^{2} - 4 \cdot (\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.4

Add parentheses.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{{(- 2 \ln (2) + 2 \ln (3))}^{2} - 4 \cdot ((\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.5

Let $u = (\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))$ . Substitute $u$ for all occurrences of $(\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))$ .

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Step 13.5.1

Rewrite ${(- 2 \ln (2) + 2 \ln (3))}^{2}$ as $(- 2 \ln (2) + 2 \ln (3)) (- 2 \ln (2) + 2 \ln (3))$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{(- 2 \ln (2) + 2 \ln (3)) (- 2 \ln (2) + 2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.2

Expand $(- 2 \ln (2) + 2 \ln (3)) (- 2 \ln (2) + 2 \ln (3))$ using the FOIL Method.

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Step 13.5.2.1

Apply the distributive property.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \ln (2) (- 2 \ln (2) + 2 \ln (3)) + 2 \ln (3) (- 2 \ln (2) + 2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.2.2

Apply the distributive property.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \ln (2) (- 2 \ln (2)) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2) + 2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.2.3

Apply the distributive property.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \ln (2) (- 2 \ln (2)) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3

Simplify and combine like terms.

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Step 13.5.3.1

Simplify each term.

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Step 13.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \cdot (- 2 \ln (2) \ln (2)) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.2

Multiply $\ln (2)$ by $\ln (2)$ by adding the exponents.

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Step 13.5.3.1.2.1

Move $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \cdot (- 2 (\ln (2) \ln (2))) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.2.2

Multiply $\ln (2)$ by $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{- 2 \cdot (- 2 \ln^{2} (2)) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.3

Multiply $- 2$ by $- 2$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 2 \ln (2) (2 \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 2 \cdot (2 \ln (2) \ln (3)) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.5

Multiply $- 2$ by $2$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) + 2 \ln (3) (- 2 \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.6

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) + 2 \cdot (- 2 \ln (3) \ln (2)) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.7

Multiply $2$ by $- 2$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) - 4 \ln (3) \ln (2) + 2 \ln (3) (2 \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) - 4 \ln (3) \ln (2) + 2 \cdot (2 \ln (3) \ln (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.9

Multiply $\ln (3)$ by $\ln (3)$ by adding the exponents.

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Step 13.5.3.1.9.1

Move $\ln (3)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) - 4 \ln (3) \ln (2) + 2 \cdot (2 (\ln (3) \ln (3))) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.9.2

Multiply $\ln (3)$ by $\ln (3)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) - 4 \ln (3) \ln (2) + 2 \cdot (2 \ln^{2} (3)) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.1.10

Multiply $2$ by $2$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (2) \ln (3) - 4 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.2

Subtract $4 \ln (3) \ln (2)$ from $- 4 \ln (2) \ln (3)$ .

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Step 13.5.3.2.1

Move $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 4 \ln (3) \ln (2) - 4 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

Step 13.5.3.2.2

Subtract $4 \ln (3) \ln (2)$ from $- 4 \ln (3) \ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 8 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 \cdot u}}{2 (\ln (2) - \ln (3))}$

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 8 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 u}}{2 (\ln (2) - \ln (3))}$

Step 13.6

Factor $4$ out of $4 \ln^{2} (2) - 8 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 u$ .

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Step 13.6.1

Factor $4$ out of $4 \ln^{2} (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) - 8 \ln (3) \ln (2) + 4 \ln^{2} (3) - 4 u}}{2 (\ln (2) - \ln (3))}$

Step 13.6.2

Factor $4$ out of $- 8 \ln (3) \ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) + 4 (- 2 \ln (3) \ln (2)) + 4 \ln^{2} (3) - 4 u}}{2 (\ln (2) - \ln (3))}$

Step 13.6.3

Factor $4$ out of $4 \ln^{2} (3)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) + 4 (- 2 \ln (3) \ln (2)) + 4 \ln^{2} (3) - 4 u}}{2 (\ln (2) - \ln (3))}$

Step 13.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 \ln^{2} (2) + 4 (- 2 \ln (3) \ln (2)) + 4 \ln^{2} (3) + 4 (- u)}}{2 (\ln (2) - \ln (3))}$

Step 13.6.5

Factor $4$ out of $4 \ln^{2} (2) + 4 (- 2 \ln (3) \ln (2))$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2)) + 4 \ln^{2} (3) + 4 (- u)}}{2 (\ln (2) - \ln (3))}$

Step 13.6.6

Factor $4$ out of $4 (\ln^{2} (2) - 2 \ln (3) \ln (2)) + 4 \ln^{2} (3)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3)) + 4 (- u)}}{2 (\ln (2) - \ln (3))}$

Step 13.6.7

Factor $4$ out of $4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3)) + 4 (- u)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - u)}}{2 (\ln (2) - \ln (3))}$

Step 13.7

Replace all occurrences of $u$ with $(\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - ((\ln (2) - \ln (3)) \cdot (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8

Simplify.

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Step 13.8.1

Simplify each term.

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Step 13.8.1.1

Expand $(\ln (2) - \ln (3)) (2 \ln (2) + \ln (\frac{3}{2}) - 2 \ln (3))$ by multiplying each term in the first expression by each term in the second expression.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (\ln (2) (2 \ln (2)) + \ln (2) \ln (\frac{3}{2}) + \ln (2) (- 2 \ln (3)) - \ln (3) (2 \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2

Simplify each term.

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Step 13.8.1.2.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln (2) \ln (2) + \ln (2) \ln (\frac{3}{2}) + \ln (2) (- 2 \ln (3)) - \ln (3) (2 \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.2

Multiply $\ln (2)$ by $\ln (2)$ by adding the exponents.

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Step 13.8.1.2.2.1

Move $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 (\ln (2) \ln (2)) + \ln (2) \ln (\frac{3}{2}) + \ln (2) (- 2 \ln (3)) - \ln (3) (2 \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.2.2

Multiply $\ln (2)$ by $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) + \ln (2) (- 2 \ln (3)) - \ln (3) (2 \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.3

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) - 2 \ln (2) \ln (3) - \ln (3) (2 \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.4

Rewrite using the commutative property of multiplication.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) - 2 \ln (2) \ln (3) - 1 \cdot (2 \ln (3) \ln (2)) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.5

Multiply $- 1$ by $2$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) - 2 \ln (2) \ln (3) - 2 \ln (3) \ln (2) - \ln (3) \ln (\frac{3}{2}) - \ln (3) (- 2 \ln (3))))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.2.6

Rewrite using the commutative property of multiplication.

Step 13.8.1.2.7

Multiply $\ln (3)$ by $\ln (3)$ by adding the exponents.

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Step 13.8.1.2.7.1

Move $\ln (3)$ .

Step 13.8.1.2.7.2

Multiply $\ln (3)$ by $\ln (3)$ .

Step 13.8.1.2.8

Multiply $- 1$ by $- 2$ .

Step 13.8.1.3

Subtract $2 \ln (3) \ln (2)$ from $- 2 \ln (2) \ln (3)$ .

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Step 13.8.1.3.1

Move $\ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) - 2 \ln (3) \ln (2) - 2 \ln (3) \ln (2) - \ln (3) \ln (\frac{3}{2}) + 2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.3.2

Subtract $2 \ln (3) \ln (2)$ from $- 2 \ln (3) \ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2) + \ln (2) \ln (\frac{3}{2}) - 4 \ln (3) \ln (2) - \ln (3) \ln (\frac{3}{2}) + 2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.4

Apply the distributive property.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - (2 \ln^{2} (2)) - (\ln (2) \ln (\frac{3}{2})) - (- 4 \ln (3) \ln (2)) - (- \ln (3) \ln (\frac{3}{2})) - (2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.5

Simplify.

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Step 13.8.1.5.1

Multiply $2$ by $- 1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - (\ln (2) \ln (\frac{3}{2})) - (- 4 \ln (3) \ln (2)) - (- \ln (3) \ln (\frac{3}{2})) - (2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.5.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - \ln (2) \ln (\frac{3}{2}) + 4 (\ln (3) \ln (2)) - (- \ln (3) \ln (\frac{3}{2})) - (2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.5.3

Multiply $- (- \ln (3) \ln (\frac{3}{2}))$ .

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Step 13.8.1.5.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - \ln (2) \ln (\frac{3}{2}) + 4 (\ln (3) \ln (2)) + 1 (\ln (3) \ln (\frac{3}{2})) - (2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.5.3.2

Multiply $\ln (3)$ by $1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - \ln (2) \ln (\frac{3}{2}) + 4 (\ln (3) \ln (2)) + \ln (3) \ln (\frac{3}{2}) - (2 \ln^{2} (3)))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.1.5.4

Multiply $2$ by $- 1$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - \ln (2) \ln (\frac{3}{2}) + 4 (\ln (3) \ln (2)) + \ln (3) \ln (\frac{3}{2}) - 2 \ln^{2} (3))}}{2 (\ln (2) - \ln (3))}$

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (\ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - 2 \ln^{2} (2) - \ln (2) \ln (\frac{3}{2}) + 4 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2}) - 2 \ln^{2} (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.2

Subtract $2 \ln^{2} (2)$ from $\ln^{2} (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (- \ln^{2} (2) - 2 \ln (3) \ln (2) + \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 4 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2}) - 2 \ln^{2} (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.3

Add $- 2 \ln (3) \ln (2)$ and $4 \ln (3) \ln (2)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (- \ln^{2} (2) + \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2}) - 2 \ln^{2} (3))}}{2 (\ln (2) - \ln (3))}$

Step 13.8.4

Subtract $2 \ln^{2} (3)$ from $\ln^{2} (3)$ .

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{4 (- \ln^{2} (2) - \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2}))}}{2 (\ln (2) - \ln (3))}$

Step 13.9

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

$x = \frac{2 \ln (2) - 2 \ln (3) \pm \sqrt{2^{2} (- \ln^{2} (2) - \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2}))}}{2 (\ln (2) - \ln (3))}$

Step 13.10

Pull terms out from under the radical.

$x = \frac{2 \ln (2) - 2 \ln (3) \pm 2 \sqrt{- \ln^{2} (2) - \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2})}}{2 (\ln (2) - \ln (3))}$

Step 14

The final answer is the combination of both solutions.

$x = \frac{\ln (2) - \ln (3) + \sqrt{- \ln^{2} (2) - \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2})}}{\ln (2) - \ln (3)}, \frac{\ln (2) - \ln (3) - \sqrt{- \ln^{2} (2) - \ln^{2} (3) - \ln (2) \ln (\frac{3}{2}) + 2 \ln (3) \ln (2) + \ln (3) \ln (\frac{3}{2})}}{\ln (2) - \ln (3)}$

Step 15

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

$x = 1, 1$