Find the discriminant of the equation and the nature of roots. Also find the roots. $6x^2 + x - 2 = 0$
- $D=49$, Real and distinct roots: $\displaystyle \frac{1}{5}, \frac{-2}{3}$
- $D=39$, Real and distinct roots: $\displaystyle \frac{1}{2}, \frac{-2}{3}$
- $D=49$, Real and distinct roots: $\displaystyle \frac{1}{3}, \frac{-7}{3}$
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$D=49$, Real and distinct roots: $\displaystyle \frac{1}{2}, \frac{-2}{3}$
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Correct answer
Explanation
D = b^2 - 4ac = 1^2 - 4(6)(-2) = 1 + 48 = 49. Roots = (-b +/- sqrt(D)) / 2a = (-1 +/- 7) / 12. Roots are 6/12 = 1/2 and -8/12 = -2/3.
AI explanation
Using the discriminant formula D equals b squared minus 4ac, we substitute a equals 6, b equals 1, and c equals negative 2. This yields D equals 1 squared minus 4 times 6 times negative 2, resulting in D equals 49. Because the discriminant is positive, the roots are real and distinct. Applying the quadratic formula gives x equals negative 1 plus 7 all divided by 12, which is one half, and x equals negative 1 minus 7 all divided by 12, which is negative two thirds.