“Using the Average” method of solving Quadratic Equations II
Yesterday I posted about having written about what Prof Po-Shen Loh recently discovered as a “different method of solving quadratic equations”. I would like to stress that I used it only in factoring while Dr Loh used it to directly determine the roots of all types of quadratics and even to derive the quadratic formula.
I would like to give more examples on how I applied this “Using the Average” method in factoring and later discuss Dr. Loh’s videos.
Factoring is often the easiest way to solve quadratics equations. However, if often involves what Dr. Loh calls “guess and check”. For quadratic equations in the form of x2 + Bx + C = 0 we guess which factor pairs of C to use and check it they will sum up to B.
In “Using the Average”, we do not have to guess. We know that the sum of the factors of C is B and their average is B/2. If d is their difference from their average, then they are (a + d) and (a -d). Their product (a + d)(a -d) or a2 – d2 is equal to C. We can therefore solve for d, and hence know the correct factor pair of C.
A few examples with clarify the method. In the example we used in “Algebra Made Easy as Arithmetic, x2 + 28x + 192 = 0, both B and C are positive. We will now consider an example where both B and C are negative
Example: x2 – 6x – 216 = 0
Here a = -6/2 = -3 and C = – 216
a2 – d2 = C
(-3)2 – d2 = – 216
9 – d2 = – 216
d2 = 9 + 216 = 225
d = ± 15
The factors are ( – 3 + 15) and (-3 – 15) or 12 and -18
X2 – 6x – 216 = (x + 12)(x – 18)
We will now consider a case when B is negative while C is positive.
Example: x2 – 15x + 36 = 0
a = -15/2 and C = 36
(-15/2)2 – d2 = 36
d2 = 225/4 – 36 = 225/4 – 144/4 = 81/4
d = 9/2
the factors of C are (a + d) = (- 15/2 + 9/2) = -6/2 = -3
and (a – d) = (- 15/2 – 9/2) = -24/2 = – 12
x2 – 15x + 36 = (x – 12)(x – 3)
We will now consider a case when the coefficient of x2 ≠ 1.
Example: 6x2 + 25x – 9 = 0
By dividing all terms by 6, this equation can be converted into x2 + 25/6 x – 9/6 = 0
Now a = (25/6)/2 = 25/12 and C = -9/6
(25/12)2 – d2 = – 9/6
d2 = 625/144 + 9/6 = 625/144 + 216/144 = 841/144
d = 29/12
(a + d) = (25/12 + 29/12) = 54/12 = 9/2
(a – d) = (25/12 – 29/12) = -4/12 = -1/3
x2 + 25/6 x – 9/6 = (x + 9/2)(x – 1/3)
If we multiply both sides of the equation by 6 then we will have
[x2 + 25/6 x – 9/6] = [ (2)(x + 9/2) (3)(x – 1/3)
or 6x2 + 25x – 9 = (2x + 9)(3x – 1)
We will discuss Dr. Loh’s video in our next post.
