Chapter 3 · Complex Numbers and Equations · Concept 2
3-8 Relationship between Roots and Coefficients (2)
This time the question tells you something about how the two roots relate to each other — and asks you to work backwards to the missing number in the equation. There is one move that starts every single one of these.
Guiding question
If you know how the two roots relate, what does that tell you about the equation?
Learning outcome. Find relationships between the roots of a quadratic and its coefficients, and use them.
Name one, and write the other in terms of it
When a question tells you how the two roots relate, let one of them be α, and then write the second one in terms of α.
After that, nothing new happens. You write down α + β = −b/a and αβ = c/a exactly as in lesson 3-7, and you now have two equations. One of them usually has no unknown constant in it — solve that one first.
Step through each of these. The naming in step 1 is the whole difficulty — after that it is last lesson's work.
the question says
the difference between them is 5
for x² − (m − 1)x − 6 = 0
step 1 · name them
the roots are ? and ?
step 2 · sum and product, from lesson 3-7
two equations appear here
step 3 · solve the one with only α in it
one of the two has no unknown constant in it
step 4 · put each answer back into the other equation
and the constant falls out
The difference between the two solutions of x² − (m − 1)x − 6 = 0 is 5. Find m and the two solutions.
- 1
Name the roots.
let them be α and α + 5The difference is 5, so the second is 5 more than the first. Nothing is being assumed — α just stands for whichever is smaller.
Work with a partner. Before solving, each of you writes down what the two roots should be called. If you have written different things — α and 3α versus α and α/3 — check whether they actually lead to the same answer. They should.
x² + ax + bc = 0 and x² + bx + ca = 0 share a root that is not zero, and a ≠ b. Show that their other two roots are the roots of x² + cx + ab = 0, where c ≠ 0.
Reporting only one branch.
When the α equation is a quadratic it has two answers, and each gives a different constant. Both belong in your answer.
Using the wrong equation first.
Pick the one with no unknown constant in it. Sometimes that is the sum, sometimes the product — look before you choose.
Adding when the question multiplies.
"3 times the other" is 3α, not α + 3. And "the square of the other" is α², not 2α.
Forgetting a on the outside.
For 8x² − mx + 1 = 0 the product is 1/8 and the sum is m/8. Neither is just 1 or m.
Throwing away a negative branch because it "looks wrong".
In Exercise 6, α = −3 gives roots −3 and 9. Check it against the question before deleting anything — it is correct.
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