[AM_20151226EIQR] Looking for a Pea among Quadratic Roots?

Question

Introduction

     This question is about finding the parameter  p, and not about solving for the “unknown”  x.  It is heavy on algebra, one has to be patient, careful and meticulous.  Please refer to this article for a recapitulation of (Vieta’s) theory of Quadratic Roots.

Solution

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve quadratic roots
* any learner who is interested

[AM_20151224QERI] Quadratic Roots and Use of Identities

Problem

Introduction

     Here is a fairly standard question on roots of quadratic equations, except that part (iii) is slightly more challenging.  To solve this question, one must know the square of sum identity well.

Recapitulation

     Please refer to this previous article  and  this article  for the theory on quadratic roots.

Solution

Heuristics Used

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

* GCE ‘A’ Levels H2 Mathematics

* other syllabuses that involve roots of quadratic equations

* any learner who is willing to learn

[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!

HeuristicsUsed

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

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve quadratic roots

* whoever loves roots and enjoy being satisfied by conquering mathematical challenges J

[IB-HL H&H_Rev6C Q11] Factors of a Complex Polynomial

Question

Introduction

     This question is taken from the Haese & Harris textbook for IB HL Mathematics and is rather challenging.  Here I present two solutions.  In the first solution, I use substitution to make a variable “disappear”.  [ heuristics: H10. Simplify the problem, H11. Solve part of the problem]  And that allowed me to crack the rest of them problem.  In the second solution, I rephrased the problem in terms of tangents to curves. [ heuristic H09. Restate the problem in another way ] This gives another angle from which to tackle the problem.

Solution 1

Solution 2

Thinking Back

     Both solutions are related in the sense that they hinge on some form of the relation maked as [*].  That led to an equation in  a.  Once a  is found,  k  can be found, and the solutions proceed similarly.  The equation [*] is a manifestation of the fact that for a repeated root, both P(x) and P’(x) share a common factor.