Answer :
To solve this problem, let's start by finding the roots of the quadratic equation [tex]\(2x^2 - 3x - 6 = 0\)[/tex]. The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -6\)[/tex].
1. Calculate the Discriminant: The discriminant of a quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[ \Delta = (-3)^2 - 4 \cdot 2 \cdot (-6) \][/tex]
[tex]\[ \Delta = 9 + 48 \][/tex]
[tex]\[ \Delta = 57 \][/tex]
So, the discriminant is [tex]\(57\)[/tex].
2. Find the Roots using the Quadratic Formula: The quadratic formula to find the roots of [tex]\(ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(\Delta = 57\)[/tex]:
[tex]\[ \alpha = \frac{-(-3) + \sqrt{57}}{2 \cdot 2} = \frac{3 + \sqrt{57}}{4} \][/tex]
[tex]\[ \beta = \frac{-(-3) - \sqrt{57}}{2 \cdot 2} = \frac{3 - \sqrt{57}}{4} \][/tex]
Thus, the roots are:
[tex]\[ \alpha = \frac{3 + \sqrt{57}}{4}, \quad \beta = \frac{3 - \sqrt{57}}{4} \][/tex]
3. Calculate [tex]\((\alpha - \beta)^2\)[/tex]:
[tex]\[ (\alpha - \beta)^2 = \left( \frac{3 + \sqrt{57}}{4} - \frac{3 - \sqrt{57}}{4} \right)^2 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ \alpha - \beta = \frac{3 + \sqrt{57}}{4} - \frac{3 - \sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{(3 + \sqrt{57}) - (3 - \sqrt{57})}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{3 + \sqrt{57} - 3 + \sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{2\sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{\sqrt{57}}{2} \][/tex]
Now square this result:
[tex]\[ (\alpha - \beta)^2 = \left(\frac{\sqrt{57}}{2}\right)^2 = \frac{57}{4} \][/tex]
So, the value of [tex]\((\alpha - \beta)^2\)[/tex] is [tex]\(\frac{57}{4}\)[/tex].
The answer is:
A. [tex]\(\frac{57}{4}\)[/tex]
1. Calculate the Discriminant: The discriminant of a quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[ \Delta = (-3)^2 - 4 \cdot 2 \cdot (-6) \][/tex]
[tex]\[ \Delta = 9 + 48 \][/tex]
[tex]\[ \Delta = 57 \][/tex]
So, the discriminant is [tex]\(57\)[/tex].
2. Find the Roots using the Quadratic Formula: The quadratic formula to find the roots of [tex]\(ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(\Delta = 57\)[/tex]:
[tex]\[ \alpha = \frac{-(-3) + \sqrt{57}}{2 \cdot 2} = \frac{3 + \sqrt{57}}{4} \][/tex]
[tex]\[ \beta = \frac{-(-3) - \sqrt{57}}{2 \cdot 2} = \frac{3 - \sqrt{57}}{4} \][/tex]
Thus, the roots are:
[tex]\[ \alpha = \frac{3 + \sqrt{57}}{4}, \quad \beta = \frac{3 - \sqrt{57}}{4} \][/tex]
3. Calculate [tex]\((\alpha - \beta)^2\)[/tex]:
[tex]\[ (\alpha - \beta)^2 = \left( \frac{3 + \sqrt{57}}{4} - \frac{3 - \sqrt{57}}{4} \right)^2 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ \alpha - \beta = \frac{3 + \sqrt{57}}{4} - \frac{3 - \sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{(3 + \sqrt{57}) - (3 - \sqrt{57})}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{3 + \sqrt{57} - 3 + \sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{2\sqrt{57}}{4} \][/tex]
[tex]\[ \alpha - \beta = \frac{\sqrt{57}}{2} \][/tex]
Now square this result:
[tex]\[ (\alpha - \beta)^2 = \left(\frac{\sqrt{57}}{2}\right)^2 = \frac{57}{4} \][/tex]
So, the value of [tex]\((\alpha - \beta)^2\)[/tex] is [tex]\(\frac{57}{4}\)[/tex].
The answer is:
A. [tex]\(\frac{57}{4}\)[/tex]
