Quadratic equations can only be solved by the method of splitting the middle term when they are written in the standard form. A quadratic equation is said to be in its standard form when the left hand side expression is a quadratic expression in the standard or general form, and is equated with zero on the right
The left hand side expression of such a quadratic equation is factored into a product of two linear algebraic expressions by the method of splitting its middle term. Then, by applying the Zero Product Rule the roots of
the quadratic equation are obtained.
Thus to solve any quadratic equation by splitting the middle term, these general steps are followed:
- Convert the quadratic equation to the standard form
- Factor the left hand side (ie, LHS) by the method of splitting the middle term
- Obtain the roots by applying the Zero Product Rule
Understanding with an example
Let us take an example to understand it clearly,
Solve: x2 + 4x = -4
Step 1: Convert the given equation to its standard form
x2 + 4x + 4 = 0
Step 2: Factor the LHS by the method of splitting the middle term
The middle term is 4x and the master product is 4x2. Thus, the two parts are 2x and 2x, because 2x times 2x is 4x2 and 2x + 2x equals 4x. Thus, the LHS expression becomes,
x2 + 2x + 2x + 4
- Form two groups: (x2 + 2x) + (2x + 4)
- Take common factors from each group: x(x + 2) + 2(x + 2)
- Take common factor from both groups: (x + 2)(x + 2)
Step 3: Obtain the roots by applying the Zero Product Rule
By the Zero Product Rule, either (x + 2) = 0 or (x + 2) = 0, and in both, x = -2.
Hence the roots of the given quadratic equation are -2 and -2.