Precalculus Examples

$3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \frac{\log_{2} (25)}{2} = 3$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{2} (25)$ as $\log_{2} (5^{2})$ .

$3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \frac{\log_{2} (5^{2})}{2} = 3$

Step 1.2

Expand $\log_{2} (5^{2})$ by moving $2$ outside the logarithm.

$3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \frac{2 \log_{2} (5)}{2} = 3$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Cancel the common factor.

$3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \frac{2 \log_{2} (5)}{2} = 3$

Step 1.3.2

Divide $\log_{2} (5)$ by $1$ .

$3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \log_{2} (5) = 3$

Step 2

Simplify the left side.

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Step 2.1

Simplify $3 \log_{2} (\sqrt{x}) - \log_{2} (x + 1) + \log_{2} (5)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $3 \log_{2} (\sqrt{x})$ by moving $3$ inside the logarithm.

$\log_{2} ({\sqrt{x}}^{3}) - \log_{2} (x + 1) + \log_{2} (5) = 3$

Step 2.1.1.2

Rewrite ${\sqrt{x}}^{3}$ as $\sqrt{x^{3}}$ .

$\log_{2} (\sqrt{x^{3}}) - \log_{2} (x + 1) + \log_{2} (5) = 3$

Step 2.1.1.3

Factor out $x^{2}$ .

$\log_{2} (\sqrt{x^{2} x}) - \log_{2} (x + 1) + \log_{2} (5) = 3$

Step 2.1.1.4

Pull terms out from under the radical.

$\log_{2} (x \sqrt{x}) - \log_{2} (x + 1) + \log_{2} (5) = 3$

Step 2.1.2

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

$\log_{2} (\frac{x \sqrt{x}}{x + 1}) + \log_{2} (5) = 3$

Step 2.1.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{x \sqrt{x}}{x + 1} \cdot 5) = 3$

Step 2.1.4

Combine $\frac{x \sqrt{x}}{x + 1}$ and $5$ .

$\log_{2} (\frac{x \sqrt{x} \cdot 5}{x + 1}) = 3$

Step 2.1.5

Move $5$ to the left of $x \sqrt{x}$ .

$\log_{2} (\frac{5 x \sqrt{x}}{x + 1}) = 3$

Step 3

Rewrite $\log_{2} (\frac{5 x \sqrt{x}}{x + 1}) = 3$ 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 $b^{y} = x$ .

$2^{3} = \frac{5 x \sqrt{x}}{x + 1}$

Step 4

Cross multiply to remove the fraction.

$5 x \sqrt{x} = 2^{3} (x + 1)$

Step 5

Simplify $2^{3} (x + 1)$ .

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Step 5.1

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

$5 x \sqrt{x} = 8 (x + 1)$

Step 5.2

Apply the distributive property.

$5 x \sqrt{x} = 8 x + 8 \cdot 1$

Step 5.3

Multiply $8$ by $1$ .

$5 x \sqrt{x} = 8 x + 8$

Step 6

Subtract $8 x$ from both sides of the equation.

$5 x \sqrt{x} - 8 x = 8$

Step 7

Factor $x$ out of $5 x \sqrt{x} - 8 x$ .

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Step 7.1

Factor $x$ out of $5 x \sqrt{x}$ .

$x (5 \sqrt{x}) - 8 x = 8$

Step 7.2

Factor $x$ out of $- 8 x$ .

$x (5 \sqrt{x}) + x \cdot - 8 = 8$

Step 7.3

Factor $x$ out of $x (5 \sqrt{x}) + x \cdot - 8$ .

$x (5 \sqrt{x} - 8) = 8$

Step 8

Solve for $5 \sqrt{x}$ .

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Step 8.1

Divide each term in $x (5 \sqrt{x} - 8) = 8$ by $x$ and simplify.

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Step 8.1.1

Divide each term in $x (5 \sqrt{x} - 8) = 8$ by $x$ .

$\frac{x (5 \sqrt{x} - 8)}{x} = \frac{8}{x}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Cancel the common factor of $x$ .

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Step 8.1.2.1.1

Cancel the common factor.

$\frac{x (5 \sqrt{x} - 8)}{x} = \frac{8}{x}$

Step 8.1.2.1.2

Divide $5 \sqrt{x} - 8$ by $1$ .

$5 \sqrt{x} - 8 = \frac{8}{x}$

Step 8.2

Add $8$ to both sides of the equation.

$5 \sqrt{x} = \frac{8}{x} + 8$

Step 9

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

${(5 \sqrt{x})}^{2} = {(\frac{8}{x} + 8)}^{2}$

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

${(5 x^{\frac{1}{2}})}^{2} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Simplify ${(5 x^{\frac{1}{2}})}^{2}$ .

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Step 10.2.1.1

Apply the product rule to $5 x^{\frac{1}{2}}$ .

$5^{2} {(x^{\frac{1}{2}})}^{2} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2.1.2

Raise $5$ to the power of $2$ .

$25 {(x^{\frac{1}{2}})}^{2} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2.1.3

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

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Step 10.2.1.3.1

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

$25 x^{\frac{1}{2} \cdot 2} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2.1.3.2

Cancel the common factor of $2$ .

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Step 10.2.1.3.2.1

Cancel the common factor.

$25 x^{\frac{1}{2} \cdot 2} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2.1.3.2.2

Rewrite the expression.

$25 x^{1} = {(\frac{8}{x} + 8)}^{2}$

Step 10.2.1.4

Simplify.

$25 x = {(\frac{8}{x} + 8)}^{2}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Simplify ${(\frac{8}{x} + 8)}^{2}$ .

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Step 10.3.1.1

Rewrite ${(\frac{8}{x} + 8)}^{2}$ as $(\frac{8}{x} + 8) (\frac{8}{x} + 8)$ .

$25 x = (\frac{8}{x} + 8) (\frac{8}{x} + 8)$

Step 10.3.1.2

Expand $(\frac{8}{x} + 8) (\frac{8}{x} + 8)$ using the FOIL Method.

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Step 10.3.1.2.1

Apply the distributive property.

$25 x = \frac{8}{x} (\frac{8}{x} + 8) + 8 (\frac{8}{x} + 8)$

Step 10.3.1.2.2

Apply the distributive property.

$25 x = \frac{8}{x} \cdot \frac{8}{x} + \frac{8}{x} \cdot 8 + 8 (\frac{8}{x} + 8)$

Step 10.3.1.2.3

Apply the distributive property.

$25 x = \frac{8}{x} \cdot \frac{8}{x} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

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 $\frac{8}{x} \cdot \frac{8}{x}$ .

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Step 10.3.1.3.1.1.1

Multiply $\frac{8}{x}$ by $\frac{8}{x}$ .

$25 x = \frac{8 \cdot 8}{x \cdot x} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.1.2

Multiply $8$ by $8$ .

$25 x = \frac{64}{x \cdot x} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.1.3

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

$25 x = \frac{64}{x^{1} x} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.1.4

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

$25 x = \frac{64}{x^{1} x^{1}} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.1.5

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

$25 x = \frac{64}{x^{1 + 1}} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.1.6

Add $1$ and $1$ .

$25 x = \frac{64}{x^{2}} + \frac{8}{x} \cdot 8 + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.2

Multiply $\frac{8}{x} \cdot 8$ .

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Step 10.3.1.3.1.2.1

Combine $\frac{8}{x}$ and $8$ .

$25 x = \frac{64}{x^{2}} + \frac{8 \cdot 8}{x} + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.2.2

Multiply $8$ by $8$ .

$25 x = \frac{64}{x^{2}} + \frac{64}{x} + 8 \frac{8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.3

Multiply $8 \frac{8}{x}$ .

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Step 10.3.1.3.1.3.1

Combine $8$ and $\frac{8}{x}$ .

$25 x = \frac{64}{x^{2}} + \frac{64}{x} + \frac{8 \cdot 8}{x} + 8 \cdot 8$

Step 10.3.1.3.1.3.2

Multiply $8$ by $8$ .

$25 x = \frac{64}{x^{2}} + \frac{64}{x} + \frac{64}{x} + 8 \cdot 8$

Step 10.3.1.3.1.4

Multiply $8$ by $8$ .

$25 x = \frac{64}{x^{2}} + \frac{64}{x} + \frac{64}{x} + 64$

Step 10.3.1.3.2

Add $\frac{64}{x}$ and $\frac{64}{x}$ .

$25 x = \frac{64}{x^{2}} + 2 \frac{64}{x} + 64$

Step 10.3.1.4

Multiply $2 \frac{64}{x}$ .

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Step 10.3.1.4.1

Combine $2$ and $\frac{64}{x}$ .

$25 x = \frac{64}{x^{2}} + \frac{2 \cdot 64}{x} + 64$

Step 10.3.1.4.2

Multiply $2$ by $64$ .

$25 x = \frac{64}{x^{2}} + \frac{128}{x} + 64$

Step 11

Solve for $x$ .

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Step 11.1

Find the LCD of the terms in the equation.

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Step 11.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, x, 1$

Step 11.1.2

Since $1, x^{2}, x, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{2}, x^{1}$ .

Step 11.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 11.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 11.1.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 11.1.6

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

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

$x$ occurs $2$ times.

Step 11.1.7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 11.1.8

The LCM of $x^{2}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 11.1.9

Multiply $x$ by $x$ .

$x^{2}$

Step 11.2

Multiply each term in $25 x = \frac{64}{x^{2}} + \frac{128}{x} + 64$ by $x^{2}$ to eliminate the fractions.

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Step 11.2.1

Multiply each term in $25 x = \frac{64}{x^{2}} + \frac{128}{x} + 64$ by $x^{2}$ .

$25 x \cdot x^{2} = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

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

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Step 11.2.2.1.1

Move $x^{2}$ .

$25 (x^{2} x) = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.2.1.2

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

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Step 11.2.2.1.2.1

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

$25 (x^{2} x^{1}) = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.2.1.2.2

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

$25 x^{2 + 1} = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.2.1.3

Add $2$ and $1$ .

$25 x^{3} = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Simplify each term.

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Step 11.2.3.1.1

Cancel the common factor of $x^{2}$ .

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Step 11.2.3.1.1.1

Cancel the common factor.

$25 x^{3} = \frac{64}{x^{2}} x^{2} + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.3.1.1.2

Rewrite the expression.

$25 x^{3} = 64 + \frac{128}{x} x^{2} + 64 x^{2}$

Step 11.2.3.1.2

Cancel the common factor of $x$ .

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Step 11.2.3.1.2.1

Factor $x$ out of $x^{2}$ .

$25 x^{3} = 64 + \frac{128}{x} (x \cdot x) + 64 x^{2}$

Step 11.2.3.1.2.2

Cancel the common factor.

$25 x^{3} = 64 + \frac{128}{x} (x \cdot x) + 64 x^{2}$

Step 11.2.3.1.2.3

Rewrite the expression.

$25 x^{3} = 64 + 128 x + 64 x^{2}$

Step 11.3

Solve the equation.

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Step 11.3.1

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

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Step 11.3.1.1

Subtract $64$ from both sides of the equation.

$25 x^{3} - 64 = 128 x + 64 x^{2}$

Step 11.3.1.2

Subtract $128 x$ from both sides of the equation.

$25 x^{3} - 64 - 128 x = 64 x^{2}$

Step 11.3.1.3

Subtract $64 x^{2}$ from both sides of the equation.

$25 x^{3} - 64 - 128 x - 64 x^{2} = 0$

Step 11.3.2

Factor the left side of the equation.

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Step 11.3.2.1

Reorder terms.

$25 x^{3} - 64 x^{2} - 128 x - 64 = 0$

Step 11.3.2.2

Factor $25 x^{3} - 64 x^{2} - 128 x - 64$ using the rational roots test.

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Step 11.3.2.2.1

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.

$p = \pm 1, \pm 64, \pm 2, \pm 32, \pm 4, \pm 16, \pm 8$

$q = \pm 1, \pm 25, \pm 5$

Step 11.3.2.2.2

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

$\pm 1, \pm 0.04, \pm 0.2, \pm 64, \pm 2.56, \pm 12.8, \pm 2, \pm 0.08, \pm 0.4, \pm 32, \pm 1.28, \pm 6.4, \pm 4, \pm 0.16, \pm 0.8, \pm 16, \pm 0.64, \pm 3.2, \pm 8, \pm 0.32, \pm 1.6$

Step 11.3.2.2.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 11.3.2.2.3.1

Substitute $4$ into the polynomial.

$25 \cdot 4^{3} - 64 \cdot 4^{2} - 128 \cdot 4 - 64$

Step 11.3.2.2.3.2

Raise $4$ to the power of $3$ .

$25 \cdot 64 - 64 \cdot 4^{2} - 128 \cdot 4 - 64$

Step 11.3.2.2.3.3

Multiply $25$ by $64$ .

$1600 - 64 \cdot 4^{2} - 128 \cdot 4 - 64$

Step 11.3.2.2.3.4

Raise $4$ to the power of $2$ .

$1600 - 64 \cdot 16 - 128 \cdot 4 - 64$

Step 11.3.2.2.3.5

Multiply $- 64$ by $16$ .

$1600 - 1024 - 128 \cdot 4 - 64$

Step 11.3.2.2.3.6

Subtract $1024$ from $1600$ .

$576 - 128 \cdot 4 - 64$

Step 11.3.2.2.3.7

Multiply $- 128$ by $4$ .

$576 - 512 - 64$

Step 11.3.2.2.3.8

Subtract $512$ from $576$ .

$64 - 64$

Step 11.3.2.2.3.9

Subtract $64$ from $64$ .

$0$

Step 11.3.2.2.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{25 x^{3} - 64 x^{2} - 128 x - 64}{x - 4}$

Step 11.3.2.2.5

Divide $25 x^{3} - 64 x^{2} - 128 x - 64$ by $x - 4$ .

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Step 11.3.2.2.5.1

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

$x$

$4$

$25 x^{3}$

$64 x^{2}$

$128 x$

$64$

Step 11.3.2.2.5.2

Divide the highest order term in the dividend $25 x^{3}$ by the highest order term in divisor $x$ .

					$25 x^{2}$
	$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$

Step 11.3.2.2.5.3

Multiply the new quotient term by the divisor.

				$25 x^{2}$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			+	$25 x^{3}$	-	$100 x^{2}$

Step 11.3.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{3} - 100 x^{2}$

				$25 x^{2}$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$

Step 11.3.2.2.5.5

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

				$25 x^{2}$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$

Step 11.3.2.2.5.6

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

				$25 x^{2}$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$

Step 11.3.2.2.5.7

Divide the highest order term in the dividend $36 x^{2}$ by the highest order term in divisor $x$ .

				$25 x^{2}$	+	$36 x$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$

Step 11.3.2.2.5.8

Multiply the new quotient term by the divisor.

				$25 x^{2}$	+	$36 x$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					+	$36 x^{2}$	-	$144 x$

Step 11.3.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $36 x^{2} - 144 x$

				$25 x^{2}$	+	$36 x$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$

Step 11.3.2.2.5.10

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

				$25 x^{2}$	+	$36 x$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$

Step 11.3.2.2.5.11

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

				$25 x^{2}$	+	$36 x$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$	-	$64$

Step 11.3.2.2.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$25 x^{2}$	+	$36 x$	+	$16$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$	-	$64$

Step 11.3.2.2.5.13

Multiply the new quotient term by the divisor.

				$25 x^{2}$	+	$36 x$	+	$16$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$	-	$64$
							+	$16 x$	-	$64$

Step 11.3.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 64$

				$25 x^{2}$	+	$36 x$	+	$16$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$	-	$64$
							-	$16 x$	+	$64$

Step 11.3.2.2.5.15

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

				$25 x^{2}$	+	$36 x$	+	$16$
$x$	-	$4$		$25 x^{3}$	-	$64 x^{2}$	-	$128 x$	-	$64$
			-	$25 x^{3}$	+	$100 x^{2}$
					+	$36 x^{2}$	-	$128 x$
					-	$36 x^{2}$	+	$144 x$
							+	$16 x$	-	$64$
							-	$16 x$	+	$64$
										$0$

Step 11.3.2.2.5.16

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

$25 x^{2} + 36 x + 16$

Step 11.3.2.2.6

Write $25 x^{3} - 64 x^{2} - 128 x - 64$ as a set of factors.

$(x - 4) (25 x^{2} + 36 x + 16) = 0$

Step 11.3.3

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

$x - 4 = 0$

$25 x^{2} + 36 x + 16 = 0$

Step 11.3.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 11.3.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 11.3.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 11.3.5

Set $25 x^{2} + 36 x + 16$ equal to $0$ and solve for $x$ .

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Step 11.3.5.1

Set $25 x^{2} + 36 x + 16$ equal to $0$ .

$25 x^{2} + 36 x + 16 = 0$

Step 11.3.5.2

Solve $25 x^{2} + 36 x + 16 = 0$ for $x$ .

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Step 11.3.5.2.1

Use the quadratic formula to find the solutions.

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

Step 11.3.5.2.2

Substitute the values $a = 25$ , $b = 36$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (25 \cdot 16)}}{2 \cdot 25}$

Step 11.3.5.2.3

Simplify.

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Step 11.3.5.2.3.1

Simplify the numerator.

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Step 11.3.5.2.3.1.1

Raise $36$ to the power of $2$ .

$x = \frac{- 36 \pm \sqrt{1296 - 4 \cdot 25 \cdot 16}}{2 \cdot 25}$

Step 11.3.5.2.3.1.2

Multiply $- 4 \cdot 25 \cdot 16$ .

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Step 11.3.5.2.3.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{- 36 \pm \sqrt{1296 - 100 \cdot 16}}{2 \cdot 25}$

Step 11.3.5.2.3.1.2.2

Multiply $- 100$ by $16$ .

$x = \frac{- 36 \pm \sqrt{1296 - 1600}}{2 \cdot 25}$

Step 11.3.5.2.3.1.3

Subtract $1600$ from $1296$ .

$x = \frac{- 36 \pm \sqrt{- 304}}{2 \cdot 25}$

Step 11.3.5.2.3.1.4

Rewrite $- 304$ as $- 1 (304)$ .

$x = \frac{- 36 \pm \sqrt{- 1 \cdot 304}}{2 \cdot 25}$

Step 11.3.5.2.3.1.5

Rewrite $\sqrt{- 1 (304)}$ as $\sqrt{- 1} \cdot \sqrt{304}$ .

$x = \frac{- 36 \pm \sqrt{- 1} \cdot \sqrt{304}}{2 \cdot 25}$

Step 11.3.5.2.3.1.6

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

$x = \frac{- 36 \pm i \cdot \sqrt{304}}{2 \cdot 25}$

Step 11.3.5.2.3.1.7

Rewrite $304$ as $4^{2} \cdot 19$ .

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Step 11.3.5.2.3.1.7.1

Factor $16$ out of $304$ .

$x = \frac{- 36 \pm i \cdot \sqrt{16 (19)}}{2 \cdot 25}$

Step 11.3.5.2.3.1.7.2

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

$x = \frac{- 36 \pm i \cdot \sqrt{4^{2} \cdot 19}}{2 \cdot 25}$

Step 11.3.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 36 \pm i \cdot (4 \sqrt{19})}{2 \cdot 25}$

Step 11.3.5.2.3.1.9

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

$x = \frac{- 36 \pm 4 i \sqrt{19}}{2 \cdot 25}$

Step 11.3.5.2.3.2

Multiply $2$ by $25$ .

$x = \frac{- 36 \pm 4 i \sqrt{19}}{50}$

Step 11.3.5.2.3.3

Simplify $\frac{- 36 \pm 4 i \sqrt{19}}{50}$ .

$x = \frac{- 18 \pm 2 i \sqrt{19}}{25}$

Step 11.3.5.2.4

The final answer is the combination of both solutions.

$x = - \frac{18 - 2 i \sqrt{19}}{25}, - \frac{18 + 2 i \sqrt{19}}{25}$

Step 11.3.6

The final solution is all the values that make $(x - 4) (25 x^{2} + 36 x + 16) = 0$ true.

$x = 4, - \frac{18 - 2 i \sqrt{19}}{25}, - \frac{18 + 2 i \sqrt{19}}{25}$