To solve the equation [tex]\( x = \frac{3}{x} \)[/tex], we need to eliminate the fraction. Here is a step-by-step solution:
1. Multiply both sides by [tex]\( x \)[/tex]:
[tex]\[
x \cdot x = x \cdot \frac{3}{x}
\][/tex]
This simplifies to:
[tex]\[
x^2 = 3
\][/tex]
2. Rewrite the equation in standard form:
[tex]\[
x^2 - 3 = 0
\][/tex]
3. Identify the coefficients for the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
a = 1, \quad b = 0, \quad c = -3
\][/tex]
4. Calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = 0^2 - 4(1)(-3) = 0 + 12 = 12
\][/tex]
5. Find the roots using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]:
[tex]\[
x = \frac{0 \pm \sqrt{12}}{2 \cdot 1}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{\sqrt{12}}{2} \quad \text{or} \quad x = \frac{-\sqrt{12}}{2}
\][/tex]
Note that [tex]\( \sqrt{12} = 2\sqrt{3} \)[/tex], so:
[tex]\[
x = \frac{2\sqrt{3}}{2} = \sqrt{3} \quad \text{or} \quad x = \frac{-2\sqrt{3}}{2} = -\sqrt{3}
\][/tex]
6. Thus, the solutions to the equation are:
[tex]\[
x = \sqrt{3} \quad \text{and} \quad x = -\sqrt{3}
\][/tex]
In conclusion, the discriminant [tex]\( \Delta = 12 \)[/tex], and the roots are [tex]\( x = \sqrt{3} \)[/tex] and [tex]\( x = -\sqrt{3} \)[/tex], which numerically approximate to [tex]\( 1.732 \)[/tex] and [tex]\( -1.732 \)[/tex], respectively.
