Precalculus Examples

$27 = 3^{5 x} \cdot 9^{x^{2}}$

Step 1

Rewrite the equation as $3^{5 x} \cdot 9^{x^{2}} = 27$ .

$3^{5 x} \cdot 9^{x^{2}} = 27$

Step 2

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

$3^{5 x} \cdot {(3^{2})}^{x^{2}} = 27$

Step 3

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

$3^{5 x} \cdot 3^{2 x^{2}} = 27$

Step 4

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

$3^{5 x + 2 x^{2}} = 27$

Step 5

Create equivalent expressions in the equation that all have equal bases.

$3^{5 x + 2 x^{2}} = 3^{3}$

Step 6

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

$5 x + 2 x^{2} = 3$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $3$ from both sides of the equation.

$5 x + 2 x^{2} - 3 = 0$

Step 7.2

Factor the left side of the equation.

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Step 7.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 7.2.2

Factor by grouping.

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Step 7.2.2.1

Reorder terms.

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

Step 7.2.2.2

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

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Step 7.2.2.2.1

Factor $5$ out of $5 u$ .

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

Step 7.2.2.2.2

Rewrite $5$ as $- 1$ plus $6$

$2 u^{2} + (- 1 + 6) u - 3 = 0$

Step 7.2.2.2.3

Apply the distributive property.

$2 u^{2} - 1 u + 6 u - 3 = 0$

Step 7.2.2.3

Factor out the greatest common factor from each group.

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Step 7.2.2.3.1

Group the first two terms and the last two terms.

$(2 u^{2} - 1 u) + 6 u - 3 = 0$

Step 7.2.2.3.2

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

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

Step 7.2.2.4

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

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

Step 7.2.3

Replace all occurrences of $u$ with $x$ .

$(2 x - 1) (x + 3) = 0$

Step 7.3

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

$2 x - 1 = 0$

$x + 3 = 0$

Step 7.4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.4.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 7.4.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 7.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 7.4.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 7.4.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.4.2.2.2

Simplify the left side.

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Step 7.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 7.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 7.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 7.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7.6

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

$x = \frac{1}{2}, - 3$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, - 3$

Decimal Form:

$x = 0.5, - 3$