Chapter 3 · Complex Numbers and Equations · Concept 2
3-6 Discriminant of Quadratic Equations
One small calculation, done before any solving, tells you how many answers there are and what kind they are. It is the piece of the quadratic formula that lives under the square root — and on its own it says more than the whole formula.
Guiding question
How can one number decide the shape of every answer?
Learning outcome. Decide the nature of the solutions using the discriminant, and use it to solve problems.
D = b² − 4ac
For ax2 + bx + c = 0, the expression under the square root in the formula is called the discriminant, written D.
D = b2 − 4ac
D > 0
two distinct real solutions
D = 0
a repeated solution (one real solution)
D < 0
two distinct imaginary solutions
The first two together are worth remembering as one line: when D ≥ 0 the solutions are real. Questions that say “has real solutions” mean D ≥ 0, with the equals included — and forgetting the equals loses exactly one number at each end of your answer.
Drag k slowly. The curve is y = x² + kx + 3 − k, and the solutions of the equation are the places where it crosses the x-axis. Watch the two crossings come together, touch, and disappear — and watch D pass through zero at the same moment.
the book's Warm Up 3
x2 + kx + 3 − k = 0
D = k2 − 4(3 − k) = -12.00
D < 0 — so it has two distinct imaginary solutions.
The lowest point of the curve is still above the axis, so there is nowhere for it to cross. The two solutions are still there — they just are not on this line.
The two solutions never actually disappear.
When the curve lifts clear of the axis there are still two solutions — they have simply left the number line, which is the only place this picture can draw. That is what “two distinct imaginary solutions” means, and it is why D < 0 does not mean “no solutions”.
2. Find the values of m for which x² − mx + m² − 3m − 9 = 0 has real solutions.
- 1
"Real solutions" means D is zero or more. Write that down first.
D ≥ 0Real INCLUDES the repeated case, so it is ≥ and not >. This single symbol is worth a mark.
Work with a partner. For question 2, one of you writes the condition (D < 0) before touching any algebra, and the other checks that the finished range really does give a negative D — pick a number inside it and try.
1. What kind of solutions?
1. What kind of solutions?
2. Find the constant.
For each of these, describe the solutions for every real value of the constant.
Reading D < 0 as "no solutions".
There are two. They are imaginary. The book names this as the main misconception of the lesson, and the picture above is what it looks like.
Reading D = 0 as "no solutions".
There is one — a repeated one. The curve touches the axis instead of crossing it.
Forgetting to flip the inequality.
Dividing −3m² + 12m + 36 ≥ 0 by −3 gives m² − 4m − 12 ≤ 0. The sign turns round. This is where most of the marks in this lesson are lost.
Using > when the question says "real".
"Has real solutions" is D ≥ 0, because a repeated solution is still real. The equals sign belongs there.
Losing the minus inside b².
If b = −m then b² = +m². If b = −(a + 1) then b² = (a + 1)². Squaring kills the minus every time.
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