Friday, November 6, 2015

[AM_20151105QFER] New Quadratic Equation satisfied from New Roots

Question

 The roots of the quadratic equation  2x2 – 3x + 6 = 0  are  a  and  b. (i)   Without finding the value of  a,  show that  8a4 = 18 – 45a. (ii)  Find the quadratic equation whose roots are  (a2 + 1)  and  (b 2 + 1).

Introduction
Do you know what a “root” is?  Is it like radish or ginseng?  Do you know what “satisfied” means?  Is it that nice feeling you get when you eat carrots?  Read on!
The featured problem above is modified from an original question that contained an error.  The modified part is shown in red.  I present two solutions.  The first solution uses pretty much standard theory, and I use notations  a’  and  b’  to denote the new roots  (a 2 + 1)  and  (b 2 + 1)  respectively.  For the second solution, I present an alternative working part (i), and one using the method of substitution for obtaining new equations (not usually taught in schools at the secondary level) for part (ii).  But before that let me first explain what “root” and “satisfied” means.

Recapitulation of Standard Theory

Solution 1 – Using Standard Theory

Solution 2 – Using the Method of Substitution for part (ii)

Remarks
Once again we can see that there are many ways to skin the cat, as it were.  Mathematics is not about following a fixed procedure.  There are various truths, notions and rules that are inviolable.  But other than that, you can have as much creativity as you want!

H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H11. Solve part of the problem
H13* Use Equation / write a Mathematical Sentence

Suitable Levels