Solve the given equation for x.
6 Superscript 7 x minus x squared Baseline equals 6 Superscript 12
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Part 1
xequals
  
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To solve the equation:

\[
6^{7x - x^2} = 6^{12}
\]

we can proceed as follows:

### Step 1: Set the Exponents Equal to Each Other
Since the bases on both sides are the same (base 6), we can equate the exponents:

\[
7x - x^2 = 12
\]

### Step 2: Rearrange into Standard Quadratic Form
Rewrite the equation as:

\[
-x^2 + 7x - 12 = 0
\]

To make it easier to solve, multiply both sides by -1:

\[
x^2 - 7x + 12 = 0
\]

### Step 3: Factor the Quadratic
Now, factor \(x^2 - 7x + 12\):

\[
(x - 3)(x - 4) = 0
\]

### Step 4: Solve for \(x\)
Set each factor equal to zero:

1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x - 4 = 0 \Rightarrow x = 4\)

### Solution
The values of \(x\) that satisfy the equation are:

\[
x = 3, 4
\]

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Would you like further clarification on any step?

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1. How do you solve quadratic equations by factoring?
2. Why can exponents be set equal to each other when the bases are the same?
3. What if the quadratic equation was not factorable?
4. How would you use the quadratic formula for this equation?
5. How do we handle equations with exponents when the bases are different?

**Tip:** Remember, when you have an equation with the same base on both sides, setting the exponents equal is a powerful simplification technique.

Solve the given equation for x. 6^(7x - x^2) = 6^12

Explore a step-by-step solution to solve the equation 6^(7x - x^2) = 6^12, involving exponent properties and quadratic equation factoring. This detailed guide is ideal for Grades 8-10 students learning algebraic techniques and exponent rules.

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