To solve the equation [tex]\(4^{x^2 - 6x} = 16384\)[/tex], let's follow a detailed, step-by-step approach:
1. Rewrite the equation with a common base:
We notice that 16384 can be rewritten as a power of 4:
[tex]\[
16384 = 4^7
\][/tex]
Therefore, we can rewrite the equation as:
[tex]\[
4^{x^2 - 6x} = 4^7
\][/tex]
2. Equate the exponents:
Since the bases are the same, the exponents must be equal. Thus, we can set the exponents equal to each other:
[tex]\[
x^2 - 6x = 7
\][/tex]
3. Form and solve the quadratic equation:
To solve for [tex]\(x\)[/tex], we rearrange the equation into standard quadratic form:
[tex]\[
x^2 - 6x - 7 = 0
\][/tex]
We now solve this quadratic equation using the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation, [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = -7\)[/tex]. Plugging in these values, we get:
[tex]\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-7)}}{2(1)}
\][/tex]
Simplifying the expression inside the square root and the entire formula:
[tex]\[
x = \frac{6 \pm \sqrt{36 + 28}}{2}
\][/tex]
[tex]\[
x = \frac{6 \pm \sqrt{64}}{2}
\][/tex]
[tex]\[
x = \frac{6 \pm 8}{2}
\][/tex]
4. Calculate the two possible solutions:
[tex]\[
x = \frac{6 + 8}{2} = \frac{14}{2} = 7
\][/tex]
[tex]\[
x = \frac{6 - 8}{2} = \frac{-2}{2} = -1
\][/tex]
So, the solutions to the equation [tex]\(4^{x^2 - 6x} = 16384\)[/tex] are:
[tex]\[
x = -1 \quad \text{and} \quad x = 7
\][/tex]
Thus, the correct answer to the problem is:
(A) 7 y -1
