Algebra Examples

$5^{x} - 5^{5 - x} = 150$

Step 1

Rewrite $5^{5 - x}$ as $5^{5} \cdot 5^{- x}$ .

$5^{x} - (5^{5} \cdot 5^{- x}) = 150$

Step 2

Rewrite $5^{- x}$ as exponentiation.

$5^{x} - (5^{5} \cdot {(5^{x})}^{- 1}) = 150$

Step 3

Remove parentheses.

$5^{x} - 5^{5} {(5^{x})}^{- 1} = 150$

Step 4

Substitute $u$ for $5^{x}$ .

$u - 5^{5} u^{- 1} = 150$

Step 5

Simplify each term.

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Step 5.1

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

$u - 1 \cdot (3125 u^{- 1}) = 150$

Step 5.2

Multiply $- 1$ by $3125$ .

$u - 3125 u^{- 1} = 150$

Step 5.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u - 3125 \frac{1}{u} = 150$

Step 5.4

Combine $- 3125$ and $\frac{1}{u}$ .

$u + \frac{- 3125}{u} = 150$

Step 5.5

Move the negative in front of the fraction.

$u - \frac{3125}{u} = 150$

Step 6

Solve for $u$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

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

$1, u, 1$

Step 6.1.2

The LCM of one and any expression is the expression.

$u$

Step 6.2

Multiply each term in $u - \frac{3125}{u} = 150$ by $u$ to eliminate the fractions.

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Step 6.2.1

Multiply each term in $u - \frac{3125}{u} = 150$ by $u$ .

$u \cdot u - \frac{3125}{u} u = 150 u$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} - \frac{3125}{u} u = 150 u$

Step 6.2.2.1.2

Cancel the common factor of $u$ .

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Step 6.2.2.1.2.1

Move the leading negative in $- \frac{3125}{u}$ into the numerator.

$u^{2} + \frac{- 3125}{u} u = 150 u$

Step 6.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 3125}{u} u = 150 u$

Step 6.2.2.1.2.3

Rewrite the expression.

$u^{2} - 3125 = 150 u$

Step 6.3

Solve the equation.

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Step 6.3.1

Subtract $150 u$ from both sides of the equation.

$u^{2} - 3125 - 150 u = 0$

Step 6.3.2

Use the quadratic formula to find the solutions.

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

Step 6.3.3

Substitute the values $a = 1$ , $b = - 150$ , and $c = - 3125$ into the quadratic formula and solve for $u$ .

$\frac{150 \pm \sqrt{{(- 150)}^{2} - 4 \cdot (1 \cdot - 3125)}}{2 \cdot 1}$

Step 6.3.4

Simplify.

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Step 6.3.4.1

Simplify the numerator.

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Step 6.3.4.1.1

Raise $- 150$ to the power of $2$ .

$u = \frac{150 \pm \sqrt{22500 - 4 \cdot 1 \cdot - 3125}}{2 \cdot 1}$

Step 6.3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 3125$ .

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Step 6.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{150 \pm \sqrt{22500 - 4 \cdot - 3125}}{2 \cdot 1}$

Step 6.3.4.1.2.2

Multiply $- 4$ by $- 3125$ .

$u = \frac{150 \pm \sqrt{22500 + 12500}}{2 \cdot 1}$

Step 6.3.4.1.3

Add $22500$ and $12500$ .

$u = \frac{150 \pm \sqrt{35000}}{2 \cdot 1}$

Step 6.3.4.1.4

Rewrite $35000$ as $50^{2} \cdot 14$ .

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Step 6.3.4.1.4.1

Factor $2500$ out of $35000$ .

$u = \frac{150 \pm \sqrt{2500 (14)}}{2 \cdot 1}$

Step 6.3.4.1.4.2

Rewrite $2500$ as $50^{2}$ .

$u = \frac{150 \pm \sqrt{50^{2} \cdot 14}}{2 \cdot 1}$

Step 6.3.4.1.5

Pull terms out from under the radical.

$u = \frac{150 \pm 50 \sqrt{14}}{2 \cdot 1}$

Step 6.3.4.2

Multiply $2$ by $1$ .

$u = \frac{150 \pm 50 \sqrt{14}}{2}$

Step 6.3.4.3

Simplify $\frac{150 \pm 50 \sqrt{14}}{2}$ .

$u = 75 \pm 25 \sqrt{14}$

Step 6.3.5

The final answer is the combination of both solutions.

$u = 75 + 25 \sqrt{14}, 75 - 25 \sqrt{14}$

Step 7

Substitute $75 + 25 \sqrt{14}$ for $u$ in $u = 5^{x}$ .

$75 + 25 \sqrt{14} = 5^{x}$

Step 8

Solve $75 + 25 \sqrt{14} = 5^{x}$ .

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Step 8.1

Rewrite the equation as $5^{x} = 75 + 25 \sqrt{14}$ .

$5^{x} = 75 + 25 \sqrt{14}$

Step 8.2

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

$\ln (5^{x}) = \ln (75 + 25 \sqrt{14})$

Step 8.3

Expand $\ln (5^{x})$ by moving $x$ outside the logarithm.

$x \ln (5) = \ln (75 + 25 \sqrt{14})$

Step 8.4

Divide each term in $x \ln (5) = \ln (75 + 25 \sqrt{14})$ by $\ln (5)$ and simplify.

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Step 8.4.1

Divide each term in $x \ln (5) = \ln (75 + 25 \sqrt{14})$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (75 + 25 \sqrt{14})}{\ln (5)}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $\ln (5)$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (75 + 25 \sqrt{14})}{\ln (5)}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (75 + 25 \sqrt{14})}{\ln (5)}$

Step 9

Substitute $75 - 25 \sqrt{14}$ for $u$ in $u = 5^{x}$ .

$75 - 25 \sqrt{14} = 5^{x}$

Step 10

Solve $75 - 25 \sqrt{14} = 5^{x}$ .

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Step 10.1

Rewrite the equation as $5^{x} = 75 - 25 \sqrt{14}$ .

$5^{x} = 75 - 25 \sqrt{14}$

Step 10.2

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

$\ln (5^{x}) = \ln (75 - 25 \sqrt{14})$

Step 10.3

The equation cannot be solved because $\ln (75 - 25 \sqrt{14})$ is undefined.

Undefined

Step 10.4

There is no solution for $5^{x} = 75 - 25 \sqrt{14}$

No solution

Step 11

List the solutions that makes the equation true.

$x = \frac{\ln (75 + 25 \sqrt{14})}{\ln (5)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (75 + 25 \sqrt{14})}{\ln (5)}$

Decimal Form:

$x = 3.18569705 \dots$

Step 13