Tutorial : Quadratic Equation Problem 1

I was given a mathematical problem recently. It was a basic problem but it took me a while to solve it as I had not been practicing my mathematics for a long time – thanks to the summer break! Anyway, here goes.

Given px² – p(3x – 1) + 3x – 1, and whereby the two roots of the equation are 2 and m. Both p and m are constants. Find the values of m and p.

First of all, you would have to figure out what special attribute of the quadratic equation to exploit. For this case, we would use the fact that x² – (A + B)x + AB = 0, whereby A and B are roots of the equation.

Working from px² – p(3x – 1) + 3x – 1, we divide the equation with p. We can divide the equation with p because p is a constant. Be careful about this.

px² – p(3x – 1) + 3x -1 …….. divide by p
= x² – 3x + 1 + 3/p x – 1/p …….. rearrange
= x² – 3x + 3/p x + 1 – 1/p …….. group them accordingly
= x² – x(3 – 3/p) + 1 – 1/p

Recalling the equation x² – (A + B)x + AB = 0, we can conclude that A+B = 3 – 3/p (1) and AB = 1 – 1/p (2) by comparison.

Let A = 2, B = m
Recall (1)
A+B = 3 – 3/p
2 + m = 3 – 3/p
m = 1 – 3/p …….. (3)

Recall (2)
AB = 1 – 1/p
2m = 1 – 1/p
Replace m with equation (3)
2(1 – 3/p) = 1 – 1/p
2 – 6/p = 1 – 1/p
1 = 5/p
p = 5

Therefore, we conclude that m = 2/5.

Therefore, the two roots of the equation are 2/5 and 2.

To check the validity of the calculation, we replace the value of p inside px² – p(3x – 1) + 3x – 1. We get :

5x² – 5(3x – 1) + 3x – 1
= 5x² – 15x + 5 + 3x -1
= 5x² – 12x + 4

Now, we know that the roots are 2/5 and 2. We can then conclude :

(x – 2/5)(x – 2) = 0

Expand the equation. We get :

x² – 2x – 2/5 x + 4/5 …….. multiply by 5
= 5x² – 10x – 2x + 4
= 5x² – 12x + 4

We obtain the exact equation.
Therefore, the calculations are valid.

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