To solve the quadratic equation [tex]\( 15x^2 - 10\sqrt{6}x + 10 = 0 \)[/tex], we need to follow a systematic approach. Here's the step-by-step solution:
1. Identify the coefficients:
- The coefficient [tex]\( a \)[/tex] for [tex]\( x^2 \)[/tex] is 15.
- The coefficient [tex]\( b \)[/tex] for [tex]\( x \)[/tex] is [tex]\(-10\sqrt{6}\)[/tex].
- The constant term [tex]\( c \)[/tex] is 10.
2. Compute the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
- Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-10\sqrt{6})^2 - 4 \times 15 \times 10
\][/tex]
- Calculate [tex]\( (-10\sqrt{6})^2 \)[/tex]:
[tex]\[
(-10\sqrt{6})^2 = 100 \times 6 = 600
\][/tex]
- Calculate [tex]\( 4 \times 15 \times 10 \)[/tex]:
[tex]\[
4 \times 15 \times 10 = 600
\][/tex]
- Now, find the discriminant:
[tex]\[
\Delta = 600 - 600 = 0
\][/tex]
3. Analyze the discriminant:
Since the discriminant [tex]\( \Delta = 0 \)[/tex], the quadratic equation has exactly one real root (with multiplicity 2). The roots are real and equal.
4. Calculate the root(s):
The formula for the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
- Substitute [tex]\( b = -10\sqrt{6} \)[/tex], [tex]\( \Delta = 0 \)[/tex], and [tex]\( a = 15 \)[/tex]:
[tex]\[
x = \frac{-(-10\sqrt{6}) \pm \sqrt{0}}{2 \times 15}
\][/tex]
- Simplify the expression:
[tex]\[
x = \frac{10\sqrt{6}}{30} = \frac{\sqrt{6}}{3}
\][/tex]
Therefore, the equation has a single root with multiplicity 2:
[tex]\[
x = \frac{\sqrt{6}}{3}
\][/tex]
So, for the quadratic equation [tex]\( 15x^2 - 10\sqrt{6}x + 10 = 0 \)[/tex], the discriminant is 0, and the sole root (with multiplicity 2) is:
[tex]\[
\boxed{x = 0.816496580927726}
\][/tex]
