For a standard quadratic equation in the form of a\(x^2\) + bx + c = 0, where a≠0, the equation has two distinct real roots when the value of the discriminant is greater than zero
For a standard quadratic equation in the form shown above, discriminant = \(b^2\) – 4ac.
Therefore, for two distinct real roots for a quadratic equation, \(b^2\) – 4ac > 0.
In questions on Quadratic equations, it is always good practice to compare the given equation to the standard form and write down the values of a, b and c. This will always help you avoid silly mistakes, because you will be able to carefully consider the values of a, b and c with due respect to their signs.
Comparing the given equation to the standard form, we have,
a = 9, b = 6k + 12 and c = \(k^2\) – 8
Substituting these values in the expression for the discriminant, \(b^2\) – 4ac, we have,
\(b^2\) – 4ac = \((6k + 12)^2\) – 4 * 9 * (\(k^2\) – 8)
Expanding using standard identities and simplifying, we have,
\(b^2\) – 4ac = 36\(k^2\) + 144 + 144k – 36\(k^2\) + 288, which further simplifies to,
\(b^2\) – 4ac = 144k + 432
Since the condition for two distinct real roots is \(b^2\) – 4ac > 0, we substitute the value of the discriminant into this standard inequality and solve. Notice how the question asks you the least value of k, as though it already knows that you will land up with an inequality.
Therefore, 144k + 432 > 0.
This means, k > - 3
Since we want the least integer value of k that is greater than -3, that value has to be -2.
The correct answer option is B.