Algebra Examples

$9^{x + \frac{1}{2}} - 4 \cdot 3^{x} + 1 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $9^{x + \frac{1}{2}}$ as $9^{x} \cdot 9^{\frac{1}{2}}$ .

$9^{x} \cdot 9^{\frac{1}{2}} - 4 \cdot 3^{x} + 1 = 0$

Step 1.2

Rewrite $9^{x}$ as ${(3^{2})}^{x}$ .

${(3^{2})}^{x} \cdot 9^{\frac{1}{2}} - 4 \cdot 3^{x} + 1 = 0$

Step 1.3

Rewrite ${(3^{2})}^{x}$ as ${(3^{x})}^{2}$ .

${(3^{x})}^{2} \cdot 9^{\frac{1}{2}} - 4 \cdot 3^{x} + 1 = 0$

Step 1.4

Let $u = 3^{x}$ . Substitute $u$ for all occurrences of $3^{x}$ .

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Step 1.4.1

Rewrite $9$ as $3^{2}$ .

$u^{2} \cdot {(3^{2})}^{\frac{1}{2}} - 4 \cdot u + 1 = 0$

Step 1.4.2

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

$u^{2} \cdot 3^{2 (\frac{1}{2})} - 4 \cdot u + 1 = 0$

Step 1.4.3

Cancel the common factor of $2$ .

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Step 1.4.3.1

Cancel the common factor.

$u^{2} \cdot 3^{2 (\frac{1}{2})} - 4 \cdot u + 1 = 0$

Step 1.4.3.2

Rewrite the expression.

$u^{2} \cdot 3 - 4 \cdot u + 1 = 0$

Step 1.4.4

Evaluate the exponent.

$u^{2} \cdot 3 - 4 \cdot u + 1 = 0$

Step 1.4.5

Move $3$ to the left of $u^{2}$ .

$3 u^{2} - 4 u + 1 = 0$

Step 1.5

Factor by grouping.

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Step 1.5.1

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 $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = - 4$ .

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Step 1.5.1.1

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

$3 u^{2} - 4 u + 1 = 0$

Step 1.5.1.2

Rewrite $- 4$ as $- 1$ plus $- 3$

$3 u^{2} + (- 1 - 3) u + 1 = 0$

Step 1.5.1.3

Apply the distributive property.

$3 u^{2} - 1 u - 3 u + 1 = 0$

Step 1.5.2

Factor out the greatest common factor from each group.

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Step 1.5.2.1

Group the first two terms and the last two terms.

$(3 u^{2} - 1 u) - 3 u + 1 = 0$

Step 1.5.2.2

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

$u (3 u - 1) - (3 u - 1) = 0$

Step 1.5.3

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

$(3 u - 1) (u - 1) = 0$

Step 1.6

Replace all occurrences of $u$ with $3^{x}$ .

$(3 \cdot 3^{x} - 1) (3^{x} - 1) = 0$

Step 2

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

$3 \cdot 3^{x} - 1 = 0$

$3^{x} - 1 = 0$

Step 3

Set $3 \cdot 3^{x} - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 \cdot 3^{x} - 1$ equal to $0$ .

$3 \cdot 3^{x} - 1 = 0$

Step 3.2

Solve $3 \cdot 3^{x} - 1 = 0$ for $x$ .

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Step 3.2.1

Multiply $3$ by $3^{x}$ .

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Step 3.2.1.1

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

$3^{1} \cdot 3^{x} - 1 = 0$

Step 3.2.1.2

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

$3^{1 + x} - 1 = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$3^{1 + x} = 1$

Step 3.2.3

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

$\ln (3^{1 + x}) = \ln (1)$

Step 3.2.4

Expand $\ln (3^{1 + x})$ by moving $1 + x$ outside the logarithm.

$(1 + x) \ln (3) = \ln (1)$

Step 3.2.5

Simplify the left side.

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Step 3.2.5.1

Simplify $(1 + x) \ln (3)$ .

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Step 3.2.5.1.1

Apply the distributive property.

$1 \ln (3) + x \ln (3) = \ln (1)$

Step 3.2.5.1.2

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

$\ln (3) + x \ln (3) = \ln (1)$

Step 3.2.6

Simplify the right side.

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Step 3.2.6.1

The natural logarithm of $1$ is $0$ .

$\ln (3) + x \ln (3) = 0$

Step 3.2.7

Reorder $\ln (3)$ and $x \ln (3)$ .

$x \ln (3) + \ln (3) = 0$

Step 3.2.8

Subtract $\ln (3)$ from both sides of the equation.

$x \ln (3) = - \ln (3)$

Step 3.2.9

Divide each term in $x \ln (3) = - \ln (3)$ by $\ln (3)$ and simplify.

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Step 3.2.9.1

Divide each term in $x \ln (3) = - \ln (3)$ by $\ln (3)$ .

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

Step 3.2.9.2

Simplify the left side.

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Step 3.2.9.2.1

Cancel the common factor of $\ln (3)$ .

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Step 3.2.9.2.1.1

Cancel the common factor.

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

Step 3.2.9.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.9.3

Simplify the right side.

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Step 3.2.9.3.1

Cancel the common factor of $\ln (3)$ .

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Step 3.2.9.3.1.1

Cancel the common factor.

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

Step 3.2.9.3.1.2

Divide $- 1$ by $1$ .

$x = - 1$

Step 4

Set $3^{x} - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3^{x} - 1$ equal to $0$ .

$3^{x} - 1 = 0$

Step 4.2

Solve $3^{x} - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$3^{x} = 1$

Step 4.2.2

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

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

Step 4.2.3

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

$x \ln (3) = \ln (1)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The natural logarithm of $1$ is $0$ .

$x \ln (3) = 0$

Step 4.2.5

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ and simplify.

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Step 4.2.5.1

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 4.2.5.2

Simplify the left side.

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Step 4.2.5.2.1

Cancel the common factor of $\ln (3)$ .

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Step 4.2.5.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 4.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{\ln (3)}$

Step 4.2.5.3

Simplify the right side.

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Step 4.2.5.3.1

Divide $0$ by $\ln (3)$ .

$x = 0$

Step 5

The final solution is all the values that make $(3 \cdot 3^{x} - 1) (3^{x} - 1) = 0$ true.

$x = - 1, 0$