Two numbers cubed add up to four,
Both real gold as you’ll see.
Their reciprocals sum to minus one,
What could those numbers be?
Two numbers cubed add up to four,
Both real gold as you’ll see.
Their reciprocals sum to minus one,
What could those numbers be?
Topics: Starter
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More Mathematics Lesson Starters
Clive, Bangkok
Saturday, March 7, 2015
"What a great starter puzzle this is. It revises so many concepts for IB students (Standard and Studies). This is how it went in my class today: The mission is to find the gold, it is in the labyrinth (of your mind!). The first obstacle is to know what kind of Maths topic this is: algebra? OK what kind of algebra is it? Simultaneous equations? Very good, can you write down those simultaneous equations? Can you solve those simultaneous equations? The simultaneous equation solver on the calculator is no good as it only solves for linear equations. How about drawing graphs and seeing where the graphs intersect? You will need to know how to rearrange the equations to make y the subject. Wow, what a surprising answer!"
Mollie Shoe, Georgia
Tuesday, May 5, 2020
"Hello, I've spent a very long time on this question and I still don't know the answer. How would you go about figuring it out?"
Pirenz, Philippines
Wednesday, January 13, 2021
"Let x and y represent the two real numbers.
1/x + 1/y = -1 {Second Given}
y + x = -xy {Multiply by xy}
Let u equal the value of x + y and -xy.
Hence:
4 = x^3 + y^3 {Given}
= (x + y) (x^2 - xy + y^2) {Factorize}
= (x + y) (x^2 + 2xy + y^2 - 3xy) {Rewrite -xy as 2xy - 3xy}
= (x + y) (x + y)^2 + 3(-xy) {Factor x^2 + 2xy + y^2}
= u (u^2 + 3u) {Substitute u = x + y and u = -xy}
= u^3 + 3u^2 {Expand}
So:
u^3 + 3u^2 - 4 = 0
(u - 1)(u + 2)(u + 2) = 0 {Factorize}
Thus, u = 1 or u = -2.
CASE I: u = -2
By definition, this means that x + y = -2 and -xy = -2
So, x + y = -2 and xy = 2
Hence, x ( -2 - x ) = 2 which means 0 = x^2 + 2x + 2.
But, x^2 + 2x + 2 is always at least 1 if x is a real number
So, this case provides no real solutions
CASE II: u = 1
By definition, this means that x + y = 1 and -xy = 1
So, x + y = 1 and xy = -1
Hence, x ( 1 - x ) = -1 which means 0 = x^2 - x - 1
Using the quadratic formula we get these answers:
x = (1 + sqrt(5))/2 and y = (1 - sqrt(5))/2
OR
x = (1 - sqrt(5))/2 and y = (1 + sqrt(5))/2
So, the numbers are (1 + sqrt(5))/2 and (1 - sqrt(5))/2
Note that (1 + sqrt(5))/2 is the golden ratio this is exactly why the second line said "Both real GOLD as you'll see".
Good one, Transum! :)."
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