Sure, let's solve the given quadratic equation step-by-step.
Given the quadratic equation:
[tex]\[ 2x^2 - 3x + 7 = 0 \][/tex]
The coefficients are:
[tex]\[ a = 2, \, b = -3, \, c = 7 \][/tex]
### Step 1: Compute the Discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the given values:
[tex]\[ \Delta = (-3)^2 - 4 \cdot 2 \cdot 7 \][/tex]
[tex]\[ \Delta = 9 - 56 \][/tex]
[tex]\[ \Delta = -47 \][/tex]
### Step 2: Analyze the Discriminant
Since the discriminant is negative ([tex]\( \Delta = -47 \)[/tex]), this indicates that the quadratic equation has no real roots. Instead, there are two complex conjugate roots.
### Step 3: Find the Complex Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\( \Delta \)[/tex] is negative, we use [tex]\( \sqrt{-\Delta} \)[/tex] and [tex]\( i \)[/tex] (the imaginary unit where [tex]\( i^2 = -1 \)[/tex]):
[tex]\[ x = \frac{-b \pm \sqrt{-47}}{2a} \][/tex]
[tex]\[ x = \frac{-(-3) \pm \sqrt{47}i}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{47}i}{4} \][/tex]
Thus, the roots are:
[tex]\[ x_1 = \frac{3 + \sqrt{47}i}{4} \][/tex]
[tex]\[ x_2 = \frac{3 - \sqrt{47}i}{4} \][/tex]
These are the complex roots of the quadratic equation [tex]\( 2x^2 - 3x + 7 = 0 \)[/tex].
### Summary
- Discriminant: [tex]\( \Delta = -47 \)[/tex]
- Roots: [tex]\( x_1 = \frac{3 + \sqrt{47}i}{4}, \, x_2 = \frac{3 - \sqrt{47}i}{4} \)[/tex]
