Solving equations and inequalities: Quadratic equations
Factorization of a quadratic equation by inspection
When a quadratic equation is in factored form, you can read off the solutions. This is based on the rule that is equivalent with or .
We discuss the method of factorisation by inspection for solving a quadratic equation of the form
for certain numbers , , and with . Because we can always divide both sides of the equation buy , it suffices to consider the special case .
Factorisation by inspection Suppose
for certain and . Then finding the solution of the original equation is easy, namely or . In this case, the following equality should be valid:
Therefore, the task has become to find two numbers and such that
The task actually has not become much easier, but sometimes you are lucky (in case of small integral coefficients) and you can see the solutions right in front of your eyes. But when it does not work, you do not really know if it is lack of inspiration or that there is indeed no solution possible in the real numbers. In that case, there is not much else to do than finding zeros via other methods and techniques.
Because of the task to find numbers and such that and, this method is also called the sum-product-method or product-sum-method. The last name indicates that the search for such numbers starts with finding pairs of integers with the prescribed product.
The given quadratic equation can be written as
You are looking for numbers and such that .
Expansion of brackets on the right-hand hand gives:
and comply with the desired properties.
The factorisation is as follows:
So, the given quadratic equation is equivalent with
and the solutions are therefore
You are looking for numbers and such that .
Expansion of brackets on the right-hand hand gives:
.
So you are looking for numbers and such that and . and comply with the desired properties.
The factorisation is as follows:
.
So, the given quadratic equation is equivalent with
and the solutions are therefore
Mathcentre video
Factorisation of a Quadratic Equation by Inspection (42:36)
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