Find the roots of the quadratic equation 6x2 – x – 2 = 0.
Step-by-Step Solution
Key Concept: Apply the quadratic formula $x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ for a quadratic equation $ax^{2}+bx+c=0$.
Given equation: $$6x^{2} - x - 2 = 0$$
1. Identify coefficients: $a = 6$, $b = -1$, $c = -2$.
2. Compute the discriminant $D$:
$$D = b^{2} - 4ac = (-1)^{2} - 4\times 6 \times (-2) = 1 + 48 = 49.$$
Since $D > 0$, the equation has two distinct real roots.
3. Apply the quadratic formula:
$$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-(-1) \pm \sqrt{49}}{2\times 6} = \frac{1 \pm 7}{12}.$$
4. Evaluate the two possibilities:
- For the ‘+’ sign:
$$x_{1} = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3}.$$
- For the ‘–’ sign:
$$x_{2} = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2}.$$
Thus, the roots of the quadratic equation are $x = \dfrac{2}{3}$ and $x = -\dfrac{1}{2}$.
Correct Answer: The roots are $x = \dfrac{2}{3}$ and $x = -\dfrac{1}{2}$.