Chapter 3 · Complex Numbers and Equations · Concept 2
3-5 Solutions of Quadratic Equations
Biologists model how a population changes from one generation to the next. At the balance point the equation is a quadratic — and whether its answers are real decides whether the population settles down or collapses.
Guiding question
How can quadratic equations tell us whether a system will stay steady or fall apart?
Learning outcome. Solve quadratic equations in the set of complex numbers.
Three ways to solve one
1 · square roots
x2 = a gives x = ±√a
(x + m)2 = a gives x = −m ± √a
2 · factorising
x(x − a) = 0 → x = 0 or a
(x − a)(x − b) = 0 → x = a or b
(x − a)2 = 0 → x = a
(ax − b)(cx − d) = 0 → x = ba or dc
3 · the formula
For ax2 + bx + c = 0
x = −b ± √(b² − 4ac)2a
Always works. Never fails. Just slower.
And the rule from lesson 3-4 still applies.
The moment a negative appears inside a square root, write it in terms of i before doing anything else. That is the only step that makes this chapter different from Grade X.
All three methods are correct on every question. But the shape of an equation quietly tells you which one will be quickest, and reading that shape is worth more marks than any of the algebra. Pick a method for each equation below and see.
the equation
x² + 8 = 0
Which way in?
(c) Solve 2x² + 2√6x + 11 = 0.
- 1
Name the three numbers before you write anything else.
a = 2, b = 2√6, c = 11Half of all formula mistakes are made here, before any calculating happens.
Work with a partner. Before solving anything, each of you writes down which method you would use and why. Compare. If you disagree, that equation is worth talking about.
1. Solve.
2. Solve.
Two of these hide something.
- · 2(e) is x² − 8x + 15 = 0, which factorises to (x − 3)(x − 5). No i anywhere. Not every equation in this chapter is complex.
- · 2(h) is 6x² − 12x + 15 = 0. Divide the whole thing by 3 first and it becomes 2x² − 4x + 5 = 0, which is far less painful.
A research team studies the vibration of a thin membrane in a medical device that uses ultrasound. The rate of energy transfer R depends on the vibration amplitude x:
R = 3x2 + 2√3x + 2
Find the values of x that make the system reach equilibrium — that is, make R zero — and explain what the answer means.
A quadratic with complex roots has no real answer. So why is that useful to a biologist studying whether a population survives?
Think about what a real root would have meant.
Squaring a surd term wrongly.
(2√6)² is 4 × 6 = 24, not 6 and not 12. The 2 in front gets squared too.
Leaving √(−64) alone until the end.
Turn it into 8i the moment it appears. Carrying a negative under a root for three more lines is how signs get lost.
Forgetting the ± .
A quadratic has two solutions. Writing only one loses half the marks, even if the one you wrote is right.
Not cancelling at the end.
(−2√6 ± 8i) ÷ 4 should be (−√6 ± 4i) ÷ 2. Check whether EVERY term on top shares a factor with the bottom.
Using the formula on (x − 3)² = −8.
It works, but you have to expand first and you will probably make a sign error doing it. Take the root of both sides instead.
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