No Real Roots: Finding a Range of k

“No real roots” is the $b^2 - 4ac < 0$ case, the third of the three discriminant conditions (alongside equal roots $= 0$ and two distinct roots $> 0$). It gives a bounded range between two values.

The equation $x^2 + kx + 9 = 0$ has no real roots. Find the range of values of the constant $k$. [4]

The working

Step 1, identify $a$, $b$, $c$: $a = 1, \qquad b = k, \qquad c = 9$

Step 2, apply the no-real-roots condition $b^2 - 4ac < 0$: $k^2 - 4(1)(9) < 0 \quad \text{(M1)}$

Step 3, simplify: $k^2 - 36 < 0 \quad \text{(A1)}$

Step 4, solve the inequality. $k^2 - 36 = (k - 6)(k + 6)$, an upward parabola in $k$, negative between its roots: $-6 < k < 6 \quad \text{(M1, A1)}$

Where the marks are won and lost

• No real roots is "$< 0$". Using $> 0$ (distinct) or $= 0$ (equal) answers a different question. Decode the phrase first.
• $k^2 - 36 < 0$ solves to the between region $-6 < k < 6$, a single bounded interval (contrast the “distinct roots” case, which gives the outside regions).
• $b = k$, so $b^2 = k^2$; keep the whole coefficient.

Common mistakes

• Using the wrong discriminant sign for “no real roots”.
• Giving the outside regions ($k < -6$ or $k > 6$) instead of the between region.
• Writing $b^2 = 2k$ or similar instead of $k^2$.

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