Basic Math Examples

$3 \cdot 3^{2 y} - 4 \cdot 3^{y} = - 1$

Step 1

Rewrite $3^{2 y}$ as exponentiation.

$3 \cdot {(3^{y})}^{2} - 4 \cdot 3^{y} = - 1$

Step 2

Substitute $u$ for $3^{y}$ .

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

Step 3

Multiply $- 4$ by $u$ .

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

Step 4

Solve for $u$ .

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Step 4.1

Add $1$ to both sides of the equation.

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

Step 4.2

Factor by grouping.

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Step 4.2.1

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 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Apply the distributive property.

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

Step 4.2.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

$3 u - 1 = 0$

$u - 1 = 0$

Step 4.4

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

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Step 4.4.1

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

$3 u - 1 = 0$

Step 4.4.2

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

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Step 4.4.2.1

Add $1$ to both sides of the equation.

$3 u = 1$

Step 4.4.2.2

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

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Step 4.4.2.2.1

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

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

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.2.2.1.2

Divide $u$ by $1$ .

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

Step 4.5

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

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Step 4.5.1

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

$u - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$u = 1$

Step 4.6

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

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

Step 5

Substitute $\frac{1}{3}$ for $u$ in $u = 3^{y}$ .

$\frac{1}{3} = 3^{y}$

Step 6

Solve $\frac{1}{3} = 3^{y}$ .

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Step 6.1

Rewrite the equation as $3^{y} = \frac{1}{3}$ .

$3^{y} = \frac{1}{3}$

Step 6.2

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

$3^{y} = \frac{1}{3^{1}}$

Step 6.3

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{y} = 3^{- 1}$

Step 6.4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$y = - 1$

Step 7

Substitute $1$ for $u$ in $u = 3^{y}$ .

$1 = 3^{y}$

Step 8

Solve $1 = 3^{y}$ .

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Step 8.1

Rewrite the equation as $3^{y} = 1$ .

$3^{y} = 1$

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Simplify the right side.

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Step 8.4.1

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

$y \ln (3) = 0$

Step 8.5

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

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Step 8.5.1

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

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

Step 8.5.2

Simplify the left side.

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Step 8.5.2.1

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

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Step 8.5.2.1.1

Cancel the common factor.

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

Step 8.5.2.1.2

Divide $y$ by $1$ .

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

Step 8.5.3

Simplify the right side.

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Step 8.5.3.1

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

$y = 0$

Step 9

List the solutions that makes the equation true.

$y = - 1, 0$