To solve the equation [tex]\( 2^x + \frac{1}{2^x} = 4.25 \)[/tex], let us follow these steps:
1. Understand the Equation:
The given equation is [tex]\( 2^x + \frac{1}{2^x} = 4.25 \)[/tex]. We need to find the values of [tex]\( x \)[/tex] that satisfy this equation.
2. Introduce a Substitution:
Let [tex]\( y = 2^x \)[/tex]. Therefore, [tex]\( \frac{1}{2^x} \)[/tex] can be rewritten as [tex]\( \frac{1}{y} \)[/tex]. The equation then becomes:
[tex]\[
y + \frac{1}{y} = 4.25
\][/tex]
3. Multiply through by [tex]\( y \)[/tex]:
To eliminate the fraction, multiply every term in the equation by [tex]\( y \)[/tex]:
[tex]\[
y \cdot y + y \cdot \frac{1}{y} = 4.25 \cdot y
\][/tex]
Which simplifies to:
[tex]\[
y^2 + 1 = 4.25y
\][/tex]
4. Rearrange into a Standard Quadratic Form:
Move all terms to one side to form a standard quadratic equation:
[tex]\[
y^2 - 4.25y + 1 = 0
\][/tex]
5. Solve the Quadratic Equation:
Use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -4.25 \)[/tex], and [tex]\( c = 1 \)[/tex]:
[tex]\[
y = \frac{-(-4.25) \pm \sqrt{(-4.25)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}
\][/tex]
[tex]\[
y = \frac{4.25 \pm \sqrt{18.0625 - 4}}{2}
\][/tex]
[tex]\[
y = \frac{4.25 \pm \sqrt{14.0625}}{2}
\][/tex]
[tex]\[
y = \frac{4.25 \pm 3.75}{2}
\][/tex]
This gives us two solutions for [tex]\( y \)[/tex]:
[tex]\[
y_1 = \frac{4.25 + 3.75}{2} = \frac{8}{2} = 4
\][/tex]
[tex]\[
y_2 = \frac{4.25 - 3.75}{2} = \frac{0.5}{2} = 0.25
\][/tex]
6. Back-substitute to Find [tex]\( x \)[/tex]:
Recall that [tex]\( y = 2^x \)[/tex]. Therefore, we need to solve:
[tex]\[
2^x = 4 \quad \text{and} \quad 2^x = 0.25
\][/tex]
For [tex]\( 2^x = 4 \)[/tex]:
[tex]\[
2^x = 2^2 \implies x = 2
\][/tex]
For [tex]\( 2^x = 0.25 \)[/tex]:
[tex]\[
2^x = 2^{-2} \implies x = -2
\][/tex]
7. State the Final Solution:
The solutions to the equation [tex]\( 2^x + \frac{1}{2^x} = 4.25 \)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = -2
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the given equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
