Algebra Examples

$24 e^{2 x} - 6 = 10 e^{x}$

Step 1

Subtract $10 e^{x}$ from both sides of the equation.

$24 e^{2 x} - 6 - 10 e^{x} = 0$

Step 2

Rewrite $e^{2 x}$ as exponentiation.

$24 {(e^{x})}^{2} - 6 - 10 e^{x} = 0$

Step 3

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

$24 u^{2} - 6 - 10 u = 0$

Step 4

Move $- 6$ .

$24 u^{2} - 10 u - 6 = 0$

Step 5

Solve for $u$ .

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Step 5.1

Factor the left side of the equation.

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Step 5.1.1

Factor $2$ out of $24 u^{2} - 10 u - 6$ .

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Step 5.1.1.1

Factor $2$ out of $24 u^{2}$ .

$2 (12 u^{2}) - 10 u - 6 = 0$

Step 5.1.1.2

Factor $2$ out of $- 10 u$ .

$2 (12 u^{2}) + 2 (- 5 u) - 6 = 0$

Step 5.1.1.3

Factor $2$ out of $- 6$ .

$2 (12 u^{2}) + 2 (- 5 u) + 2 (- 3) = 0$

Step 5.1.1.4

Factor $2$ out of $2 (12 u^{2}) + 2 (- 5 u)$ .

$2 (12 u^{2} - 5 u) + 2 (- 3) = 0$

Step 5.1.1.5

Factor $2$ out of $2 (12 u^{2} - 5 u) + 2 (- 3)$ .

$2 (12 u^{2} - 5 u - 3) = 0$

Step 5.1.2

Factor.

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Step 5.1.2.1

Factor by grouping.

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Step 5.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 3 = - 36$ and whose sum is $b = - 5$ .

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Step 5.1.2.1.1.1

Factor $- 5$ out of $- 5 u$ .

$2 (12 u^{2} - 5 u - 3) = 0$

Step 5.1.2.1.1.2

Rewrite $- 5$ as $4$ plus $- 9$

$2 (12 u^{2} + (4 - 9) u - 3) = 0$

Step 5.1.2.1.1.3

Apply the distributive property.

$2 (12 u^{2} + 4 u - 9 u - 3) = 0$

Step 5.1.2.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1.2.1

Group the first two terms and the last two terms.

$2 ((12 u^{2} + 4 u) - 9 u - 3) = 0$

Step 5.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 (4 u (3 u + 1) - 3 (3 u + 1)) = 0$

Step 5.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$2 ((3 u + 1) (4 u - 3)) = 0$

Step 5.1.2.2

Remove unnecessary parentheses.

$2 (3 u + 1) (4 u - 3) = 0$

Step 5.2

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

$3 u + 1 = 0$

$4 u - 3 = 0$

Step 5.3

Set $3 u + 1$ equal to $0$ and solve for $u$ .

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Step 5.3.1

Set $3 u + 1$ equal to $0$ .

$3 u + 1 = 0$

Step 5.3.2

Solve $3 u + 1 = 0$ for $u$ .

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Step 5.3.2.1

Subtract $1$ from both sides of the equation.

$3 u = - 1$

Step 5.3.2.2

Divide each term in $3 u = - 1$ by $3$ and simplify.

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Step 5.3.2.2.1

Divide each term in $3 u = - 1$ by $3$ .

$\frac{3 u}{3} = \frac{- 1}{3}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{- 1}{3}$

Step 5.3.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{3}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{3}$

Step 5.4

Set $4 u - 3$ equal to $0$ and solve for $u$ .

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Step 5.4.1

Set $4 u - 3$ equal to $0$ .

$4 u - 3 = 0$

Step 5.4.2

Solve $4 u - 3 = 0$ for $u$ .

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Step 5.4.2.1

Add $3$ to both sides of the equation.

$4 u = 3$

Step 5.4.2.2

Divide each term in $4 u = 3$ by $4$ and simplify.

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Step 5.4.2.2.1

Divide each term in $4 u = 3$ by $4$ .

$\frac{4 u}{4} = \frac{3}{4}$

Step 5.4.2.2.2

Simplify the left side.

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Step 5.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{3}{4}$

Step 5.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{3}{4}$

Step 5.5

The final solution is all the values that make $2 (3 u + 1) (4 u - 3) = 0$ true.

$u = - \frac{1}{3}, \frac{3}{4}$

Step 6

Substitute $- \frac{1}{3}$ for $u$ in $u = e^{x}$ .

$- \frac{1}{3} = e^{x}$

Step 7

Solve $- \frac{1}{3} = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = - \frac{1}{3}$ .

$e^{x} = - \frac{1}{3}$

Step 7.2

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

$\ln (e^{x}) = \ln (- \frac{1}{3})$

Step 7.3

The equation cannot be solved because $\ln (- \frac{1}{3})$ is undefined.

Undefined

Step 7.4

There is no solution for $e^{x} = - \frac{1}{3}$

No solution

Step 8

Substitute $\frac{3}{4}$ for $u$ in $u = e^{x}$ .

$\frac{3}{4} = e^{x}$

Step 9

Solve $\frac{3}{4} = e^{x}$ .

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Step 9.1

Rewrite the equation as $e^{x} = \frac{3}{4}$ .

$e^{x} = \frac{3}{4}$

Step 9.2

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

$\ln (e^{x}) = \ln (\frac{3}{4})$

Step 9.3

Expand the left side.

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Step 9.3.1

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

$x \ln (e) = \ln (\frac{3}{4})$

Step 9.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (\frac{3}{4})$

Step 9.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{3}{4})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{3}{4})$

Decimal Form:

$x = - 0.28768207 \dots$