Calculus Examples

${(27)}^{x + 6} = {(3)}^{x^{2}}$

Step 1

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

$3^{3 (x + 6)} = 3^{x^{2}}$

Step 2

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

$3 (x + 6) = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (x + 6)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (x + 6) = x^{2}$

Step 3.1.2

Simplify by adding zeros.

$3 (x + 6) = x^{2}$

Step 3.1.3

Apply the distributive property.

$3 x + 3 \cdot 6 = x^{2}$

Step 3.1.4

Multiply $3$ by $6$ .

$3 x + 18 = x^{2}$

Step 3.2

Subtract $x^{2}$ from both sides of the equation.

$3 x + 18 - x^{2} = 0$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $- 1$ out of $3 x + 18 - x^{2}$ .

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Step 3.3.1.1

Reorder the expression.

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Step 3.3.1.1.1

Move $18$ .

$3 x - x^{2} + 18 = 0$

Step 3.3.1.1.2

Reorder $3 x$ and $- x^{2}$ .

$- x^{2} + 3 x + 18 = 0$

Step 3.3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 3 x + 18 = 0$

Step 3.3.1.3

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) + 18 = 0$

Step 3.3.1.4

Rewrite $18$ as $- 1 (- 18)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot - 18 = 0$

Step 3.3.1.5

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot - 18 = 0$

Step 3.3.1.6

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (- 18)$ .

$- (x^{2} - 3 x - 18) = 0$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 3.3.2.1.2

Write the factored form using these integers.

$- ((x - 6) (x + 3)) = 0$

Step 3.3.2.2

Remove unnecessary parentheses.

$- (x - 6) (x + 3) = 0$

Step 3.4

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

$x - 6 = 0$

$x + 3 = 0$

Step 3.5

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.6

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

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Step 3.6.1

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

$x + 3 = 0$

Step 3.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.7

The final solution is all the values that make $- (x - 6) (x + 3) = 0$ true.

$x = 6, - 3$