To solve the equation [tex]\( 3^{2x} - 3^x = 72 \)[/tex], we'll follow a series of algebraic steps. Let's go through it step-by-step:
1. Rewrite the Equation:
Let's introduce a substitution to simplify the equation. Let [tex]\( y = 3^x \)[/tex]. The original equation [tex]\( 3^{2x} - 3^x = 72 \)[/tex] can be written in terms of [tex]\( y \)[/tex] as follows:
[tex]\[
3^{2x} = (3^x)^2 = y^2
\][/tex]
So, the equation becomes:
[tex]\[
y^2 - y = 72
\][/tex]
2. Form a Quadratic Equation:
Rewrite the simplified equation in the standard quadratic form:
[tex]\[
y^2 - y - 72 = 0
\][/tex]
3. Solve the Quadratic Equation:
To solve for [tex]\( y \)[/tex], we can use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -72 \)[/tex]:
[tex]\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1}
\][/tex]
[tex]\[
y = \frac{1 \pm \sqrt{1 + 288}}{2}
\][/tex]
[tex]\[
y = \frac{1 \pm \sqrt{289}}{2}
\][/tex]
[tex]\[
y = \frac{1 \pm 17}{2}
\][/tex]
This gives us two solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1 + 17}{2} = \frac{18}{2} = 9
\][/tex]
[tex]\[
y = \frac{1 - 17}{2} = \frac{-16}{2} = -8
\][/tex]
4. Substitute Back for [tex]\( 3^x \)[/tex]:
Recall that [tex]\( y = 3^x \)[/tex]. Therefore, we have:
[tex]\[
3^x = 9 \quad \text{or} \quad 3^x = -8
\][/tex]
Since [tex]\( 3^x \)[/tex] is always positive for real numbers [tex]\( x \)[/tex], the negative solution [tex]\( 3^x = -8 \)[/tex] cannot be valid. Thus, we consider only:
[tex]\[
3^x = 9
\][/tex]
5. Find [tex]\( x \)[/tex]:
Express 9 as a power of 3:
[tex]\[
3^x = 3^2
\][/tex]
By comparing the exponents, we get:
[tex]\[
x = 2
\][/tex]
Therefore, the solution to the equation [tex]\( 3^{2x} - 3^x = 72 \)[/tex] is [tex]\( x = 2 \)[/tex]. Thus, the correct answer is:
(a) 2
