Solve for x 6^(3x^2-3x)=36^(15x-36)

$6^{3 x^{2} - 3 x} = 36^{15 x - 36}$

Step 1

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

$6^{3 x^{2} - 3 x} = 6^{2 (15 x - 36)}$

Step 2

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

$3 x^{2} - 3 x = 2 (15 x - 36)$

Step 3

Solve for $x$ .

Simplify $2 (15 x - 36)$ .

Apply the distributive property.

$3 x^{2} - 3 x = 2 (15 x) + 2 \cdot - 36$

Multiply.

Multiply $15$ by $2$ .

$3 x^{2} - 3 x = 30 x + 2 \cdot - 36$

Multiply $2$ by $- 36$ .

$3 x^{2} - 3 x = 30 x - 72$

Move all terms containing $x$ to the left side of the equation.

Subtract $30 x$ from both sides of the equation.

$3 x^{2} - 3 x - 30 x = - 72$

Subtract $30 x$ from $- 3 x$ .

$3 x^{2} - 33 x = - 72$

Add $72$ to both sides of the equation.

$3 x^{2} - 33 x + 72 = 0$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 33 x + 72$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 33 x + 72 = 0$

Factor $3$ out of $- 33 x$ .

$3 (x^{2}) + 3 (- 11 x) + 72 = 0$

Factor $3$ out of $72$ .

$3 x^{2} + 3 (- 11 x) + 3 \cdot 24 = 0$

Factor $3$ out of $3 x^{2} + 3 (- 11 x)$ .

$3 (x^{2} - 11 x) + 3 \cdot 24 = 0$

Factor $3$ out of $3 (x^{2} - 11 x) + 3 \cdot 24$ .

$3 (x^{2} - 11 x + 24) = 0$

Factor.

Factor $x^{2} - 11 x + 24$ using the AC method.

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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Write the factored form using these integers.

$3 ((x - 8) (x - 3)) = 0$

Remove unnecessary parentheses.

$3 (x - 8) (x - 3) = 0$

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

$x - 8 = 0$

$x - 3 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $3 (x - 8) (x - 3) = 0$ true.

$x = 8, 3$

Solve for x 6^(3x^2-3x)=36^(15x-36)

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