Precalculus Examples

$\log (8 x) - \log (1 + \sqrt{x}) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{8 x}{1 + \sqrt{x}}) = 2$

Step 1.2

Multiply $\frac{8 x}{1 + \sqrt{x}}$ by $\frac{1 - \sqrt{x}}{1 - \sqrt{x}}$ .

$\log (\frac{8 x}{1 + \sqrt{x}} \cdot \frac{1 - \sqrt{x}}{1 - \sqrt{x}}) = 2$

Step 1.3

Multiply $\frac{8 x}{1 + \sqrt{x}}$ by $\frac{1 - \sqrt{x}}{1 - \sqrt{x}}$ .

$\log (\frac{8 x (1 - \sqrt{x})}{(1 + \sqrt{x}) (1 - \sqrt{x})}) = 2$

Step 1.4

Expand the denominator using the FOIL method.

$\log (\frac{8 x (1 - \sqrt{x})}{1 - \sqrt{x} + \sqrt{x} - {\sqrt{x}}^{2}}) = 2$

Step 1.5

Simplify.

$\log (\frac{8 x (1 - \sqrt{x})}{- x + 1}) = 2$

Step 2

Rewrite $\log (\frac{8 x (1 - \sqrt{x})}{- x + 1}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{2} = \frac{8 x (1 - \sqrt{x})}{- x + 1}$

Step 3

Cross multiply to remove the fraction.

$8 x (1 - \sqrt{x}) = 10^{2} (- x + 1)$

Step 4

Simplify $10^{2} (- x + 1)$ .

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Step 4.1

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

$8 x (1 - \sqrt{x}) = 100 (- x + 1)$

Step 4.2

Apply the distributive property.

$8 x (1 - \sqrt{x}) = 100 (- x) + 100 \cdot 1$

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $100$ .

$8 x (1 - \sqrt{x}) = - 100 x + 100 \cdot 1$

Step 4.3.2

Multiply $100$ by $1$ .

$8 x (1 - \sqrt{x}) = - 100 x + 100$

Step 5

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

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Step 5.1

Add $100 x$ to both sides of the equation.

$8 x (1 - \sqrt{x}) + 100 x = 100$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$8 x \cdot 1 + 8 x (- \sqrt{x}) + 100 x = 100$

Step 5.2.2

Multiply $8$ by $1$ .

$8 x + 8 x (- \sqrt{x}) + 100 x = 100$

Step 5.2.3

Multiply $- 1$ by $8$ .

$8 x - 8 x \sqrt{x} + 100 x = 100$

Step 5.3

Add $8 x$ and $100 x$ .

$- 8 x \sqrt{x} + 108 x = 100$

Step 6

Factor $4 x$ out of $- 8 x \sqrt{x} + 108 x$ .

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Step 6.1

Factor $4 x$ out of $- 8 x \sqrt{x}$ .

$4 x (- 2 \sqrt{x}) + 108 x = 100$

Step 6.2

Factor $4 x$ out of $108 x$ .

$4 x (- 2 \sqrt{x}) + 4 x (27) = 100$

Step 6.3

Factor $4 x$ out of $4 x (- 2 \sqrt{x}) + 4 x (27)$ .

$4 x (- 2 \sqrt{x} + 27) = 100$

Step 7

Divide each term in $4 x (- 2 \sqrt{x} + 27) = 100$ by $4$ and simplify.

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Step 7.1

Divide each term in $4 x (- 2 \sqrt{x} + 27) = 100$ by $4$ .

$\frac{4 x (- 2 \sqrt{x} + 27)}{4} = \frac{100}{4}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $4$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{4 x (- 2 \sqrt{x} + 27)}{4} = \frac{100}{4}$

Step 7.2.1.2

Divide $x (- 2 \sqrt{x} + 27)$ by $1$ .

$x (- 2 \sqrt{x} + 27) = \frac{100}{4}$

Step 7.2.2

Apply the distributive property.

$x (- 2 \sqrt{x}) + x \cdot 27 = \frac{100}{4}$

Step 7.2.3

Reorder.

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Step 7.2.3.1

Rewrite using the commutative property of multiplication.

$- 2 x \sqrt{x} + x \cdot 27 = \frac{100}{4}$

Step 7.2.3.2

Move $27$ to the left of $x$ .

$- 2 x \sqrt{x} + 27 x = \frac{100}{4}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $100$ by $4$ .

$- 2 x \sqrt{x} + 27 x = 25$

Step 8

Subtract $27 x$ from both sides of the equation.

$- 2 x \sqrt{x} = 25 - 27 x$

Step 9

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

${(- 2 x \sqrt{x})}^{2} = {(25 - 27 x)}^{2}$

Step 10

Simplify each side of the equation.

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Step 10.1

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

${(- 2 x \cdot x^{\frac{1}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2

Simplify the left side.

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Step 10.2.1

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

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Step 10.2.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 10.2.1.1.1

Move $x^{\frac{1}{2}}$ .

${(- 2 (x^{\frac{1}{2}} x))}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.1.2

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

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Step 10.2.1.1.2.1

Raise $x$ to the power of $1$ .

${(- 2 (x^{\frac{1}{2}} x^{1}))}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- 2 x^{\frac{1}{2} + 1})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.1.3

Write $1$ as a fraction with a common denominator.

${(- 2 x^{\frac{1}{2} + \frac{2}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.1.4

Combine the numerators over the common denominator.

${(- 2 x^{\frac{1 + 2}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.1.5

Add $1$ and $2$ .

${(- 2 x^{\frac{3}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.2

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

${(- 2)}^{2} {(x^{\frac{3}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.3

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

$4 {(x^{\frac{3}{2}})}^{2} = {(25 - 27 x)}^{2}$

Step 10.2.1.4

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

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Step 10.2.1.4.1

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

$4 x^{\frac{3}{2} \cdot 2} = {(25 - 27 x)}^{2}$

Step 10.2.1.4.2

Cancel the common factor of $2$ .

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Step 10.2.1.4.2.1

Cancel the common factor.

$4 x^{\frac{3}{2} \cdot 2} = {(25 - 27 x)}^{2}$

Step 10.2.1.4.2.2

Rewrite the expression.

$4 x^{3} = {(25 - 27 x)}^{2}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Simplify ${(25 - 27 x)}^{2}$ .

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Step 10.3.1.1

Rewrite ${(25 - 27 x)}^{2}$ as $(25 - 27 x) (25 - 27 x)$ .

$4 x^{3} = (25 - 27 x) (25 - 27 x)$

Step 10.3.1.2

Expand $(25 - 27 x) (25 - 27 x)$ using the FOIL Method.

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Step 10.3.1.2.1

Apply the distributive property.

$4 x^{3} = 25 (25 - 27 x) - 27 x (25 - 27 x)$

Step 10.3.1.2.2

Apply the distributive property.

$4 x^{3} = 25 \cdot 25 + 25 (- 27 x) - 27 x (25 - 27 x)$

Step 10.3.1.2.3

Apply the distributive property.

$4 x^{3} = 25 \cdot 25 + 25 (- 27 x) - 27 x \cdot 25 - 27 x (- 27 x)$

Step 10.3.1.3

Simplify and combine like terms.

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Step 10.3.1.3.1

Simplify each term.

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Step 10.3.1.3.1.1

Multiply $25$ by $25$ .

$4 x^{3} = 625 + 25 (- 27 x) - 27 x \cdot 25 - 27 x (- 27 x)$

Step 10.3.1.3.1.2

Multiply $- 27$ by $25$ .

$4 x^{3} = 625 - 675 x - 27 x \cdot 25 - 27 x (- 27 x)$

Step 10.3.1.3.1.3

Multiply $25$ by $- 27$ .

$4 x^{3} = 625 - 675 x - 675 x - 27 x (- 27 x)$

Step 10.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 x^{3} = 625 - 675 x - 675 x - 27 \cdot - 27 x \cdot x$

Step 10.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 10.3.1.3.1.5.1

Move $x$ .

$4 x^{3} = 625 - 675 x - 675 x - 27 \cdot - 27 (x \cdot x)$

Step 10.3.1.3.1.5.2

Multiply $x$ by $x$ .

$4 x^{3} = 625 - 675 x - 675 x - 27 \cdot - 27 x^{2}$

Step 10.3.1.3.1.6

Multiply $- 27$ by $- 27$ .

$4 x^{3} = 625 - 675 x - 675 x + 729 x^{2}$

Step 10.3.1.3.2

Subtract $675 x$ from $- 675 x$ .

$4 x^{3} = 625 - 1350 x + 729 x^{2}$

Step 11

Solve for $x$ .

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Step 11.1

Move all the expressions to the left side of the equation.

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Step 11.1.1

Subtract $625$ from both sides of the equation.

$4 x^{3} - 625 = - 1350 x + 729 x^{2}$

Step 11.1.2

Add $1350 x$ to both sides of the equation.

$4 x^{3} - 625 + 1350 x = 729 x^{2}$

Step 11.1.3

Subtract $729 x^{2}$ from both sides of the equation.

$4 x^{3} - 625 + 1350 x - 729 x^{2} = 0$

Step 11.2

Factor the left side of the equation.

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Step 11.2.1

Reorder terms.

$4 x^{3} - 729 x^{2} + 1350 x - 625 = 0$

Step 11.2.2

Factor $4 x^{3} - 729 x^{2} + 1350 x - 625$ using the rational roots test.

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Step 11.2.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 625, \pm 5, \pm 125, \pm 25$

$q = \pm 1, \pm 4, \pm 2$

Step 11.2.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.25, \pm 0.5, \pm 625, \pm 156.25, \pm 312.5, \pm 5, \pm 1.25, \pm 2.5, \pm 125, \pm 31.25, \pm 62.5, \pm 25, \pm 6.25, \pm 12.5$

Step 11.2.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 11.2.2.3.1

Substitute $1$ into the polynomial.

$4 \cdot 1^{3} - 729 \cdot 1^{2} + 1350 \cdot 1 - 625$

Step 11.2.2.3.2

Raise $1$ to the power of $3$ .

$4 \cdot 1 - 729 \cdot 1^{2} + 1350 \cdot 1 - 625$

Step 11.2.2.3.3

Multiply $4$ by $1$ .

$4 - 729 \cdot 1^{2} + 1350 \cdot 1 - 625$

Step 11.2.2.3.4

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

$4 - 729 \cdot 1 + 1350 \cdot 1 - 625$

Step 11.2.2.3.5

Multiply $- 729$ by $1$ .

$4 - 729 + 1350 \cdot 1 - 625$

Step 11.2.2.3.6

Subtract $729$ from $4$ .

$- 725 + 1350 \cdot 1 - 625$

Step 11.2.2.3.7

Multiply $1350$ by $1$ .

$- 725 + 1350 - 625$

Step 11.2.2.3.8

Add $- 725$ and $1350$ .

$625 - 625$

Step 11.2.2.3.9

Subtract $625$ from $625$ .

$0$

Step 11.2.2.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 729 x^{2} + 1350 x - 625}{x - 1}$

Step 11.2.2.5

Divide $4 x^{3} - 729 x^{2} + 1350 x - 625$ by $x - 1$ .

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Step 11.2.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$4 x^{3}$

$729 x^{2}$

$1350 x$

$625$

Step 11.2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

					$4 x^{2}$
	$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$

Step 11.2.2.5.3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			+	$4 x^{3}$	-	$4 x^{2}$

Step 11.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 4 x^{2}$

				$4 x^{2}$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$

Step 11.2.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$

Step 11.2.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$4 x^{2}$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$

Step 11.2.2.5.7

Divide the highest order term in the dividend $- 725 x^{2}$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$725 x$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$

Step 11.2.2.5.8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$725 x$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					-	$725 x^{2}$	+	$725 x$

Step 11.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 725 x^{2} + 725 x$

				$4 x^{2}$	-	$725 x$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$

Step 11.2.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$	-	$725 x$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$

Step 11.2.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$4 x^{2}$	-	$725 x$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$	-	$625$

Step 11.2.2.5.12

Divide the highest order term in the dividend $625 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$725 x$	+	$625$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$	-	$625$

Step 11.2.2.5.13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$725 x$	+	$625$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$	-	$625$
							+	$625 x$	-	$625$

Step 11.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $625 x - 625$

				$4 x^{2}$	-	$725 x$	+	$625$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$	-	$625$
							-	$625 x$	+	$625$

Step 11.2.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$	-	$725 x$	+	$625$
$x$	-	$1$		$4 x^{3}$	-	$729 x^{2}$	+	$1350 x$	-	$625$
			-	$4 x^{3}$	+	$4 x^{2}$
					-	$725 x^{2}$	+	$1350 x$
					+	$725 x^{2}$	-	$725 x$
							+	$625 x$	-	$625$
							-	$625 x$	+	$625$
										$0$

Step 11.2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$4 x^{2} - 725 x + 625$

Step 11.2.2.6

Write $4 x^{3} - 729 x^{2} + 1350 x - 625$ as a set of factors.

$(x - 1) (4 x^{2} - 725 x + 625) = 0$

Step 11.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$4 x^{2} - 725 x + 625 = 0$

Step 11.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 11.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 11.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 11.5

Set $4 x^{2} - 725 x + 625$ equal to $0$ and solve for $x$ .

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Step 11.5.1

Set $4 x^{2} - 725 x + 625$ equal to $0$ .

$4 x^{2} - 725 x + 625 = 0$

Step 11.5.2

Solve $4 x^{2} - 725 x + 625 = 0$ for $x$ .

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Step 11.5.2.1

Use the quadratic formula to find the solutions.

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

Step 11.5.2.2

Substitute the values $a = 4$ , $b = - 725$ , and $c = 625$ into the quadratic formula and solve for $x$ .

$\frac{725 \pm \sqrt{{(- 725)}^{2} - 4 \cdot (4 \cdot 625)}}{2 \cdot 4}$

Step 11.5.2.3

Simplify.

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Step 11.5.2.3.1

Simplify the numerator.

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Step 11.5.2.3.1.1

Raise $- 725$ to the power of $2$ .

$x = \frac{725 \pm \sqrt{525625 - 4 \cdot 4 \cdot 625}}{2 \cdot 4}$

Step 11.5.2.3.1.2

Multiply $- 4 \cdot 4 \cdot 625$ .

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Step 11.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{725 \pm \sqrt{525625 - 16 \cdot 625}}{2 \cdot 4}$

Step 11.5.2.3.1.2.2

Multiply $- 16$ by $625$ .

$x = \frac{725 \pm \sqrt{525625 - 10000}}{2 \cdot 4}$

Step 11.5.2.3.1.3

Subtract $10000$ from $525625$ .

$x = \frac{725 \pm \sqrt{515625}}{2 \cdot 4}$

Step 11.5.2.3.1.4

Rewrite $515625$ as $125^{2} \cdot 33$ .

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Step 11.5.2.3.1.4.1

Factor $15625$ out of $515625$ .

$x = \frac{725 \pm \sqrt{15625 (33)}}{2 \cdot 4}$

Step 11.5.2.3.1.4.2

Rewrite $15625$ as $125^{2}$ .

$x = \frac{725 \pm \sqrt{125^{2} \cdot 33}}{2 \cdot 4}$

Step 11.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{725 \pm 125 \sqrt{33}}{2 \cdot 4}$

Step 11.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{725 \pm 125 \sqrt{33}}{8}$

Step 11.5.2.4

The final answer is the combination of both solutions.

$x = \frac{725 + 125 \sqrt{33}}{8}, \frac{725 - 125 \sqrt{33}}{8}$

Step 11.6

The final solution is all the values that make $(x - 1) (4 x^{2} - 725 x + 625) = 0$ true.

$x = 1, \frac{725 + 125 \sqrt{33}}{8}, \frac{725 - 125 \sqrt{33}}{8}$

Step 12

Exclude the solutions that do not make $\log (8 x) - \log (1 + \sqrt{x}) = 2$ true.

$x = \frac{725 + 125 \sqrt{33}}{8}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{725 + 125 \sqrt{33}}{8}$

Decimal Form:

$x = 180.38379135 \dots$