Answer:
Either [tex]-4\sqrt{6}[/tex] or [tex]4\sqrt{6}[/tex], depending on whether [tex]\alpha[/tex] is larger than [tex]\beta[/tex].
Step-by-step explanation:
The two roots (might necessarily be distinct or real) of the quadratic equation
[tex]ax^{2} + bx + c = 0[/tex], where [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are constants and [tex]a\ne 0[/tex] are
- [tex]\displaystyle x_1 = \frac{-b+\sqrt{\text{b^{2} - 4ac}}}{2a}[/tex], and
- [tex]\displaystyle x_2 = \frac{-b-\sqrt{\text{b^{2} - 4ac}}}{2a}[/tex].
The difference between the two will be either:
[tex]x_1 - x_2 = 2\sqrt{b^{2} - 4ac}[/tex] or
[tex]x_2 - x_1 = -2\sqrt{b^{2} - 4ac}[/tex].
For this question,
- [tex]a = 3[/tex],
- [tex]b = -6[/tex], and
- [tex]c = -1[/tex].
[tex]x_1 - x_2 = 2\sqrt{(-6)^{2} - 4\times 3\times (-1)} = 4\sqrt{6}[/tex], or
[tex]x_1 - x_2 = -2\sqrt{(-6)^{2} - 4\times 3\times (-1)} = -4\sqrt{6}[/tex].
