Trigonometry Examples

$81^{x^{3} + 2 x^{2}} = 27^{\frac{5 x}{3}}$

Step 1

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

$3^{4 (x^{3} + 2 x^{2})} = 3^{3 \frac{5 x}{3}}$

Step 2

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

$4 (x^{3} + 2 x^{2}) = 3 \frac{5 x}{3}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify $4 (x^{3} + 2 x^{2})$ .

Tap for more steps...

Step 3.1.1

Rewrite.

$0 + 0 + 4 (x^{3} + 2 x^{2}) = 3 (\frac{5 x}{3})$

Step 3.1.2

Simplify by adding zeros.

$4 (x^{3} + 2 x^{2}) = 3 (\frac{5 x}{3})$

Step 3.1.3

Apply the distributive property.

$4 x^{3} + 4 (2 x^{2}) = 3 (\frac{5 x}{3})$

Step 3.1.4

Multiply $2$ by $4$ .

$4 x^{3} + 8 x^{2} = 3 (\frac{5 x}{3})$

Step 3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$4 x^{3} + 8 x^{2} = 3 \frac{5 x}{3}$

Step 3.2.2

Rewrite the expression.

$4 x^{3} + 8 x^{2} = 5 x$

Step 3.3

Subtract $5 x$ from both sides of the equation.

$4 x^{3} + 8 x^{2} - 5 x = 0$

Step 3.4

Factor the left side of the equation.

Tap for more steps...

Step 3.4.1

Factor $x$ out of $4 x^{3} + 8 x^{2} - 5 x$ .

Tap for more steps...

Step 3.4.1.1

Factor $x$ out of $4 x^{3}$ .

$x (4 x^{2}) + 8 x^{2} - 5 x = 0$

Step 3.4.1.2

Factor $x$ out of $8 x^{2}$ .

$x (4 x^{2}) + x (8 x) - 5 x = 0$

Step 3.4.1.3

Factor $x$ out of $- 5 x$ .

$x (4 x^{2}) + x (8 x) + x \cdot - 5 = 0$

Step 3.4.1.4

Factor $x$ out of $x (4 x^{2}) + x (8 x)$ .

$x (4 x^{2} + 8 x) + x \cdot - 5 = 0$

Step 3.4.1.5

Factor $x$ out of $x (4 x^{2} + 8 x) + x \cdot - 5$ .

$x (4 x^{2} + 8 x - 5) = 0$

Step 3.4.2

Factor.

Tap for more steps...

Step 3.4.2.1

Factor by grouping.

Tap for more steps...

Step 3.4.2.1.1

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 $a \cdot c = 4 \cdot - 5 = - 20$ and whose sum is $b = 8$ .

Tap for more steps...

Step 3.4.2.1.1.1

Factor $8$ out of $8 x$ .

$x (4 x^{2} + 8 (x) - 5) = 0$

Step 3.4.2.1.1.2

Rewrite $8$ as $- 2$ plus $10$

$x (4 x^{2} + (- 2 + 10) x - 5) = 0$

Step 3.4.2.1.1.3

Apply the distributive property.

$x (4 x^{2} - 2 x + 10 x - 5) = 0$

Step 3.4.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.4.2.1.2.1

Group the first two terms and the last two terms.

$x ((4 x^{2} - 2 x) + 10 x - 5) = 0$

Step 3.4.2.1.2.2

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

$x (2 x (2 x - 1) + 5 (2 x - 1)) = 0$

Step 3.4.2.1.3

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

$x ((2 x - 1) (2 x + 5)) = 0$

Step 3.4.2.2

Remove unnecessary parentheses.

$x (2 x - 1) (2 x + 5) = 0$

Step 3.5

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

$x = 0$

$2 x - 1 = 0$

$2 x + 5 = 0$

Step 3.6

Set $x$ equal to $0$ .

$x = 0$

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

$2 x - 1 = 0$

Step 3.7.2

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

Tap for more steps...

Step 3.7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.7.2.2

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

Tap for more steps...

Step 3.7.2.2.1

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

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

Step 3.7.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.7.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.7.2.2.2.1.1

Cancel the common factor.

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

Step 3.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.8

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.8.1

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Step 3.8.2

Solve $2 x + 5 = 0$ for $x$ .

Tap for more steps...

Step 3.8.2.1

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Step 3.8.2.2

Divide each term in $2 x = - 5$ by $2$ and simplify.

Tap for more steps...

Step 3.8.2.2.1

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 3.8.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.8.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.8.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 3.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Step 3.8.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.8.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 3.9

The final solution is all the values that make $x (2 x - 1) (2 x + 5) = 0$ true.

$x = 0, \frac{1}{2}, - \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{2}, - \frac{5}{2}$

Decimal Form:

$x = 0, 0.5, - 2.5$

Mixed Number Form:

$x = 0, \frac{1}{2}, - 2 \frac{1}{2}$