Theory of Intelligence
A-LEVEL•Mathematics•Quadratics•Question 12

Solution

🔴 Stretch · [5 marks] Find the set of values of the constant $k$ for which the curve $y = kx^2 + 3x + k$ lies entirely below the $x$-axis.

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Solution

For the curve to be entirely below the xx-axis it must:

  1. open downwards, so k<0k < 0, and
  2. have no real roots, so b2−4ac<0b^2 - 4ac < 0.
9−4k⋅k<04k2>9k2>94k<−32ork>32\begin{aligned} 9 - 4k \cdot k &< 0 \\ 4k^2 &> 9 \\ k^2 &> \tfrac94 \\ k < -\tfrac32 \quad&\text{or}\quad k > \tfrac32 \end{aligned}

Combining with k<0k < 0:

k<−32k < -\tfrac32

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