Chapter 3 · Complex Numbers and Equations · Concept 2
3-10 Quadratic Equations with Two Given Solutions
Lesson 3-7 took an equation apart to find the sum and product of its roots. This puts one back together: give me two numbers, and I will hand you an equation that has exactly those two as its solutions.
Guiding question
If you know the answers, can you write the question?
Learning outcome. Form a quadratic equation when its roots are known.
One line does the whole lesson
One quadratic equation with the numbers α and β as its solutions is
x2 − (α + β)x + αβ = 0
Notice the word “one”.
There is not a single answer. x² − 3x + 2 = 0 and 2x² − 6x + 4 = 0 and −5x² + 15x − 10 = 0 all have the same two roots. The question usually says “create one quadratic equation”, and sometimes adds “with integer coefficients” — which is the hint to multiply through and clear any fractions.
The sign is where the marks go. The sum goes in with a MINUS and the product with a PLUS. If the two roots are 3 and 5, the sum is 8 and the equation is x² − 8x + 15 = 0. Test it: 9 − 24 + 15 = 0 ✓
Step through each pair. Watch the sum arrive with its sign changed, and watch what happens to the i and the √7 when the two numbers are conjugates.
the two numbers
−1 and 2
their sum
?
their product
?
drop them into x² − (sum)x + (product) = 0
…
3. α and β solve x² + 2x − 4 = 0. Find a quadratic equation, with 1 in front of x², whose two solutions are α + 2 and β + 2.
- 1
Get the OLD sum and product first. Nothing else can start until these are down.
α + β = −2 and αβ = −4Straight from lesson 3-7 on x² + 2x − 4 = 0, where a = 1, b = 2, c = −4.
Work with a partner. Whoever is not writing should check the finished equation by putting one of the two numbers into it. If it does not give zero, the equation is wrong — and finding that out now costs one minute.
1. Create one quadratic equation with these two numbers as its solutions.
2. Find two numbers with these sums and products.
3–5. Build a new equation from an old one.
The figure in the book shows a parabola crossing the x-axis at α and at β — so α and β are the two roots of some quadratic. Create a quadratic equation whose roots are 1/α and 1/β.
Putting the sum in with a plus.
The rule is x² − (sum)x + (product) = 0. Roots 3 and 5 give x² − 8x + 15 = 0, not x² + 8x + 15 = 0.
Ignoring "with integer coefficients".
That phrase means multiply through to clear every fraction. x² − x + 1/2 = 0 should be handed in as 2x² − 2x + 1 = 0.
Substituting numbers for α and β.
You do not know them, and you do not need them. Work only with α + β and αβ.
Forgetting to expand before substituting.
(α + 2)(β + 2) has to become αβ + 2(α + β) + 4 first. Only then does it turn into numbers you have.
Squaring the sum to get the new product.
For roots α² and β², the new product is (αβ)², not (α + β)². The new SUM is the one that needs (α + β)² − 2αβ.
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