Certainly! Let's solve the equation [tex]\( 4^x - 20 \times 2^x + 64 = 0 \)[/tex] step-by-step.
### Step 1: Simplify [tex]\( 4^x \)[/tex]
Notice that [tex]\( 4^x \)[/tex] can be written as [tex]\( (2^2)^x \)[/tex], which simplifies to [tex]\( (2^x)^2 \)[/tex]. Let [tex]\( y = 2^x \)[/tex]. This reduces the equation to a quadratic form:
[tex]\[
(2^x)^2 - 20 \cdot 2^x + 64 = 0 \implies y^2 - 20y + 64 = 0
\][/tex]
### Step 2: Solve the Quadratic Equation
We now have a quadratic equation in terms of [tex]\( y \)[/tex]:
[tex]\[
y^2 - 20y + 64 = 0
\][/tex]
To solve this, we can use the quadratic formula, [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -20 \)[/tex], and [tex]\( c = 64 \)[/tex]:
[tex]\[
y = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}
\][/tex]
### Step 3: Calculate the Discriminant
First, let's find the discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[
\Delta = (-20)^2 - 4 \cdot 1 \cdot 64 = 400 - 256 = 144
\][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now plug the discriminant back into the quadratic formula:
[tex]\[
y = \frac{20 \pm \sqrt{144}}{2} = \frac{20 \pm 12}{2}
\][/tex]
This gives us two solutions:
[tex]\[
y = \frac{20 + 12}{2} = \frac{32}{2} = 16
\][/tex]
[tex]\[
y = \frac{20 - 12}{2} = \frac{8}{2} = 4
\][/tex]
### Step 5: Back-Substitute [tex]\( y = 2^x \)[/tex]
Recall that [tex]\( y = 2^x \)[/tex], so we have two equations to solve:
1. [tex]\( 2^x = 16 \)[/tex]
2. [tex]\( 2^x = 4 \)[/tex]
### Step 6: Solve for [tex]\( x \)[/tex]
For [tex]\( 2^x = 16 \)[/tex]:
[tex]\[
2^x = 2^4 \implies x = 4
\][/tex]
For [tex]\( 2^x = 4 \)[/tex]:
[tex]\[
2^x = 2^2 \implies x = 2
\][/tex]
### Conclusion
The values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 4^x - 20 \times 2^x + 64 = 0 \)[/tex] are:
[tex]\[
\boxed{2 \text{ and } 4}
\][/tex]
