Sure, let's solve the quadratic equation [tex]\( x^2 - x + 10 = 0 \)[/tex] step-by-step.
### 1. Identify Coefficients
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = 10 \)[/tex]
### 2. Calculate the Discriminant
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Plugging in our coefficients:
[tex]\[
\Delta = (-1)^2 - 4 \cdot 1 \cdot 10 = 1 - 40 = -39
\][/tex]
### 3. Interpret the Discriminant
The discriminant [tex]\( \Delta = -39 \)[/tex] is negative. This tells us that the equation has no real roots. Instead, it has two complex conjugate roots.
### 4. Calculate the Complex Roots
When the discriminant is negative, the roots of the quadratic equation can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\( \Delta = -39 \)[/tex], we need to use the imaginary unit [tex]\( i \)[/tex] (where [tex]\( i^2 = -1 \)[/tex]) to express the square root of the negative discriminant:
[tex]\[
\sqrt{-39} = \sqrt{39}i
\][/tex]
Now, substitute [tex]\( \Delta \)[/tex] and the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-1) \pm \sqrt{-39}}{2 \cdot 1} = \frac{1 \pm \sqrt{39}i}{2}
\][/tex]
### 5. Simplify the Expression
Thus, the roots can be written as:
[tex]\[
x = \frac{1 + \sqrt{39}i}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{39}i}{2}
\][/tex]
So the solutions to the quadratic equation [tex]\( x^2 - x + 10 = 0 \)[/tex] are:
[tex]\[
x = \frac{1 + \sqrt{39}i}{2}
\][/tex]
[tex]\[
x = \frac{1 - \sqrt{39}i}{2}
\][/tex]
These are the two complex conjugate roots of the given quadratic equation.
