The original question includes a diagram: Eight labelled grids with p on the horizontal axis and q on the vertical axis, each running from −1 to 6 and −4 to 4, showing different shaded regions bounded by dashed lines through the origin.
The graph of the quadratic y = px2 + qx + p where p > 0, intersects the x-axis at two distinct points.
In which one of the following graphs does the shaded region show the complete set of possible values that p and q could take?
Each of the eight graphs plots p on the horizontal axis and q on the vertical axis, with the boundary lines drawn dashed. They are described below.
- AEverything above the dashed line q = 2p together with the whole region below the p-axis, for p > 0.
- BOnly the triangle bounded by the q-axis, the line q = 4 and the dashed line q = 2p.
- CEverything below and to the right of the dashed line q = 2p, for p > 0.
- DOnly the triangle above the dashed line q = 2p with p > 0, drawn as far as q = 4.
- EThe single region lying to the right of both dashed lines q = 2p and q = −2p, for p > 0.
- FThe two wedges with p > 0 lying above the dashed line q = 2p and below the dashed line q = −2p.
- GEverything except the wedge between the dashed lines q = 2p and q = −2p on the right, so the shading also covers p < 0.
- HThe two wedges lying to the left of the dashed lines q = 2p and q = −2p, extending into p < 0.
Show the answer and worked solution
answer · F
- AEverything above the dashed line q = 2p together with the whole region below the p-axis, for p > 0.
- BOnly the triangle bounded by the q-axis, the line q = 4 and the dashed line q = 2p.
- CEverything below and to the right of the dashed line q = 2p, for p > 0.
- DOnly the triangle above the dashed line q = 2p with p > 0, drawn as far as q = 4.
- EThe single region lying to the right of both dashed lines q = 2p and q = −2p, for p > 0.
- FThe two wedges with p > 0 lying above the dashed line q = 2p and below the dashed line q = −2p.
- GEverything except the wedge between the dashed lines q = 2p and q = −2p on the right, so the shading also covers p < 0.
- HThe two wedges lying to the left of the dashed lines q = 2p and q = −2p, extending into p < 0.
Two distinct roots means a positive discriminant: q2 − 4 × p × p > 0, so q2 > 4p2. Since p > 0, taking square roots gives |q| > 2p, that is q > 2p or q < −2p. On the (p, q) plane that is a pair of open wedges in the right half-plane — one above the line of gradient 2 through the origin, one below the line of gradient −2 — with the lines themselves excluded because the inequality is strict. Graph F is the one showing exactly those two wedges and nothing for p ≤ 0.