# “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 **x ^{2}
+ 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** a ^{2} – d^{2}** 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, **x ^{2} + 28x +
192 = 0**, both

**B**and

**C**are positive. We will now consider an example where both B and C are negative

Example: **x ^{2}
– 6x – 216 = 0**

Here **a =
-6/2 = -3** and **C = – 216**

**a ^{2}
– d^{2} = C**

(-3)^{2}
– d^{2} = – 216

9 – d^{2}
= – 216

d^{2 }=
9 + 216 = 225

d = ± 15

The factors are ( – 3 + 15) and (-3 – 15) or 12 and -18

**X ^{2}
– 6x – 216 = (x + 12)(x – 18)**

We will now consider a case when B is negative while C is positive.

Example: **x ^{2}
– 15x + 36 = 0**

**a = -15/2
and C = 36**

(-15/2)^{2}
– d^{2} = 36

d^{2}
= 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**

**x ^{2}
– 15x + 36 = (x – 12)(x – 3)**

We will now
consider a case when the coefficient of x^{2} ≠ 1.

Example: **6x ^{2} + 25x – 9 = 0**

By dividing all terms by 6, this equation can be converted into **x ^{2} + 25/6 x – 9/6 = 0**

Now **a = (25/6)/2
= 25/12 and C = -9/6**

**(25/12) ^{2}
– d^{2} = – 9/6**

**d ^{2}** = 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**

**x ^{2}
+ 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

**[x ^{2} + 25/6 x – 9/6] = [ (2)(x + 9/2) (3)(x – 1/3)**

or **6x ^{2} + 25x – 9 = (2x + 9)(3x – 1)**

We will discuss Dr. Loh’s video in our next post.