Splitting the middle term (Additional Solved Examples)

Example 1: 3x² + 5x + 2 = 0


Solution:

Since the given qaudratic equation is already in its standard form, we don't need
to convert it to the standard form. Factor the left hand side quadratic expression
by the method of splitting the middle term:


The master product is 3x² * 2 = 6x² and the middle term is
5x. Which two algebraic terms have the product of 6x² and the sum of
5x? They are 2x and 3x. Thus, the middle term 5x is split into two parts, 2x and
3x. Hence the equation becomes

3x² + 3x + 2x + 2 = 0

Now factor the left hand side by making two groups each containing two terms. Then
factor out the common factors from each group. Hence the equation becomes

3x(x + 1) + 2(x + 1) = 0

(x + 1) is the common factor in the above expression. When the method of splitting
the middle term is done correctly on a non prime quadratic expression, there is
always a common factor between the two groups. Thus factoring out (x + 1), the equation
becomes


(x + 1)(3x + 2) = 0

The above obtained equation is now completely factored. Now we can apply the Zero
Product Rule to obtain the roots of the above equation. The Zero Product Rule states
that if the product of two quantities is zero, then either of the two quantities
is itslef equal to zero. Hence,

Either (x + 1) = 0, in which case x = -1

Or (3x + 2) = 0, in which case x = -2/3

Hence the roots of the given quadratic expression are -1 and -2/3.

Example 2: 14x - 49x² = -1


Solution:

The given quadratic equation is not in its standard form, so converting it to standard
form:


49x² - 14x + 1 = 0

Now factor the left hand side expression by splitting its middle term. The master
product of the quadratic expression is -49x² and its middle term is -14x.
Which two algebraic terms have the product of -49x² and the sum of -14x?
-7x and -7x are two terms whose product is -49x² and sum is -14x. Thus
the two parts of the middle term are -7x and -7x. Rewrite the quadratic equation
with the middle term split into the two parts:


49x² - 7x - 7x + 1 = 0


Factor the quadratic expression on the left hand side further by making two groups
- each group containing two terms - and factoring out the common factors from each
group. The resulting quadratic equation is

7x(7x - 1) - 1(7x - 1) = 0


(7x - 1) is the common factor between the two groups. Hence the result is,


(7x - 1)(7x - 1) = 0


That is,

(7x - 1)² = 0

Therefore 7x - 1 = 0, which implies that x = 1/7

Hence the given quadratic equation has two equal roots, 1/7 and 1/7.

Example 3: Find the values of 'p' in the equation 4p² + 12pq + 9q²
= 0


Solution:

The left hand side expression of this quadratic equation contains two variables.
The equation is such that it can be considered as written in either 'p' or 'q'.
But since the instructions clearly say that you have to find the values of 'p',
we will write the equation in 'p' only.




The given quadratic equation is already in the standard form with respect to the
variable 'p'. In order to factor its left hand side, split the middle term according
to the master product method.




The middle term of the quadratic expression on the left hand side is 12pq, and its
master product is 36p²q². Thus the two algebraic terms whose
product is equal to the master product and whose sum is equal to the middle term
are 6pq and 6pq. (6pq + 6pq = 12pq and 6pq * 6pq = 36p²q².
Rewriting the equation with the middle term split into these two parts, we obtain




4p² + 6pq + 6pq + 9q² = 0




Further factor the left hand side by grouping and factoring out common factors.
Thus the equation becomes,




2p(2p + 3q) + 3q(2p + 3q) = 0




Since (2p + 3q) is a common factor, factoring it out, the equation becomes




(2p + 3q)(2p + 3q) = 0




Thus 2p + 3q = 0, and hence p = -3q/2




Thus the given equation has equal two roots with respect to the variable 'p'. These
are -3q/2 and -3q/2.

Example 4: Find the solution for (a + b) for the equation: (a + b)² -
16(a + b) = -48

Solution:

In this quadratic equation, the variable is not just a single variable, rather it
is an algebraic expression : (a + b). In order to view it as a single variable quadratic
equation, let y = (a + b), then the equation can be rewritten as


y² + 16y = -48


Rewrite the equation in the standard form (ax² + bx + c = 0),

y² + 16y + 48 = 0


Split the middle term of the quadratic expression on the left hand side. The middle
term is 16y and the master product is 48y². Which two algebraic terms
have the sum of 16y and the product of 48y²? The two required algebraic
terms are 12y and 4y because 12y + 4y equals 16y and 12y times 4y equals 48y².
Thus rewriting the equation with these two terms,


y² + 4y + 12y + 48 = 0


Factor the left hand side by grouping into two groups, one group containing the
first two terms and the other group containing the last two terms,

(y² + 4y) + (12y + 48) = 0


Take common factors from each group:

y(y + 4) + 12(y + 4) = 0


Take (y + 4) as the common factor:


(y + 4)(y + 12) = 0


By applying the zero product rule, we obtain the roots of the quadratic equation,


Either y + 4 = 0 or y + 12 = 0

Either y = -4 or y = -12

Since y = a + b, therefore either (a + b) = -4 or (a + b) = -12

Therefore the roots of the given equation are -4 and -12.

Example 5: 12m² + m = 1

Solution:

Rewrite the equation in the standard form:

12m² + m - 1 = 0


In order to factor the quadratic expression, split the middle term into two
parts such that their product is equal to the product of the first (ie 12m²) and last (ie -1) terms
of the quadratic expression.


12m² + 4m - 3m - 1 = 0


Make two groups of two terms each,


(12m² + 4m) + (- 3m - 1) = 0


Factor out the common factors from each group,


4m(3m + 1) - 1(3m + 1) = 0


(3m + 1) is a common factor between the two groups, so factoring it out


(3m + 1)(4m - 1) = 0


By the Zero Product Rule, either (3m + 1) = 0 or (4m - 1) = 0

That is, either m = -1/3 or m = 1/4.

Hence the roots of the given equation are -1/3 and 1/4.