Two Distinct Roots: Finding a Range of k

The discriminant condition you use depends entirely on the wording. “Two distinct real roots” means $b^2 - 4ac > 0$, an inequality, so the answer is a range of values, not specific ones. This is the counterpart to the equal-roots question, where the discriminant is set to zero instead.

The equation $2x^2 + kx + 8 = 0$ has two distinct real roots. Find the range of values of the constant $k$. [4]

The working

Step 1, identify $a$, $b$, $c$: $a = 2, \qquad b = k, \qquad c = 8$

Step 2, apply the distinct-roots condition $b^2 - 4ac > 0$: $k^2 - 4(2)(8) > 0 \quad \text{(M1)}$

Step 3, simplify: $k^2 - 64 > 0 \quad \text{(A1)}$

Step 4, solve the inequality. $k^2 - 64 = (k - 8)(k + 8)$, an upward parabola in $k$, positive outside the roots $\pm 8$: $k < -8 \quad \text{or} \quad k > 8 \quad \text{(M1, A1)}$

Where the marks are won and lost

• Distinct roots is "$> 0$", not "$= 0$". Using $= 0$ answers the equal-roots question and gives only $k = \pm 8$ instead of the range.
• $k^2 - 64 > 0$ solves to the outside regions $k < -8$ or $k > 8$, joined by “or”. Writing $-8 < k < 8$ (the inside) is the region where there are no real roots.
• Keep $b = k$ as the whole coefficient: $b^2 = k^2$, not $k$.

Common mistakes

• Using the equal-roots condition ($= 0$) and giving $k = \pm 8$.
• Giving the “between” region instead of “outside”.
• Writing $b^2 = 2k$ or similar instead of $k^2$.

Full method: Discriminant & Nature of Roots notes. Topic home: Quadratic Functions pillar.