ADDITIONAL MATHS
“Wonderful quadratic inequalities” OR “Yes
theyʼre tricky, but weʼre persistent...”
I have consulted my wide network of maths-teacher contacts and uncovered two new methods which do not involve any graph drawing. They all rely on being able to factorise & an understanding of these 3 vital principles:
PRINCIPLE 1 If two numbers multiply to equal zero:
One, or both, numbers equal zero
PRINCIPLE 2 If two numbers multiply to equal a ‘positive’:
Both numbers are negative or both are positive
PRINCIPLE 3 If two numbers multiply to equal a ‘negative’:
One number is negative and one is positive
EXAMPLE (Method 1) Let’s take the example x2 + 2x − 8 ≥ 0 To ‘solve’ this, means “find what numbers which when put into that formula are greater than or equal to zero”. Let’s factorise it. Remember: start each bracket with an x then try to fill the brackets with numbers which multiple to give − 8. i.
(x + 4) (x − 2) ≥ 0
e. (x + 4) times (x -2) positive, is or zero...
Our principles tell us that (x + 4) and (x - 2) must be both positive or both negative (or zero). THIS IS THE KEY. EITHER x + 4 ≥ 0 and x − 2 ≥ 0 x ≥ −4 and x ≥ 2 i.e. x ≥ 2 which is the final solution.
OR x + 4 ≤ 0 and x − 2 ≤ 0 x ≤ −4 and x ≤ 2 i.e. ≤ −4
EXAMPLE (Method 2) Let’s take the example x2 + 2x − 8 ≥ 0 We’re more familiar with quadratic equations so let’s look at x2 + 2x − 8 = 0 - but the ≥ 0 bit will be very important at the end. We know how to solve this: let’s factorise it. Remember: start each bracket with an x then try to fill the brackets with numbers which multiple to give − 8.
(x + 4) (x − 2) = 0 Principle 1 tell us that either (x + 4) or (x - 2) must be zero. (They can’t both be zero because you need different values of x).
EITHER
x = −4 A
OR
x=2 B
−4
2
C
Letʼs ma rk them on the number line for now.. .
This splits the line into 3 sections. It must follow that in each part, (x + 4) (x − 2) is either positive or negative. The final part of the method is deciding which part is which. Itʼs easier to use the factorised version. Just pick any point in each section. SECTION A (x < -4): letʼs pick x = -5 (x + 4) (x − 2) = (−5 + 4) (−5 − 2) = (−1) (−7) = 7.
Greater than 0
SECTION B (-4 < x < 2): letʼs pick x = 0 (x + 4) (x − 2) = (0 + 4) (0 − 2) = (4) (−2) = −8.
Less than 0
SECTION C (x > 2): letʼs pick x = 3 (x + 4) (x − 2) = (3 + 4) (3 − 2) = (7) (1) = 7.
Greater than 0
Remember that the question wanted ≥ 0. So the answer to the question is section A and section C. Our final solution is
x < − 4 or x > 2