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Pure Quadratic Equations

Definition:
A quadratic equation in which the term containing x without an exponent is not present is called a pure quadratic equation. In other words, a quadratic equation in which the term containing x raised to the power of 1 is not present is called a pure quadratic equation.

A pure quadratic equation can also be described as a quadratic equation in which there are only two terms, one is the term containing x-squared, and one is the constant number.

Sometimes, as you would notice in the below examples, pure quadratic equations behave like difference of two squares, which can be very useful to solve them quickly.


Examples:
  • x2 = 25
  • x2 = 36
  • x2 = -81
  • x2 - 121 = 0
  • x2 - 25 = 0
  • x2 + 4 = 0
The above are six examples of pure quadratic equations. In the above examples, the fourth and fifth examples are pure quadratic equations that are in the form of difference of two squares.

Solving:
Pure quadratic equations are easy to solve. You don't need to factor a pure quadratic equation to solve it, although you can. There are two methods to solve pure quadratic equations:
  1. Taking square roots on both sides
  2. Difference of two squares method
The first method, that of taking square roots, will help you solve all types of pure quadratic equations. The second method will allow you to solve only those pure quadratic equations that are in the form of difference of two squares. Let us learn how to do each method:

Taking square roots on both sides:
Let us learn this method by solving the pure quadratic equation x2 - 50 = 0
Step 1: Move the number/constant term to the right hand side of the equation, if not already.
x2 = 50
Step 2: Take square roots on both sides
x^2 = √50
x = + √50  or x = - √50
If required, you can simplify the above radicals to get x =  5√2, x = -5√2
Difference of two squares method:
This method can only be applied when the pure quadratic equation to be solved is in the form of difference of two squares. For example, the pure quadratic x2 - 25 = 0 can also be written as x2 - 52 = 0, because 5 squared equals 25.

Applying the factorization formula for difference of two squares, we obtain:
(x + 5) (x - 5) = 0
By applying the zero product rule to obtain the roots, we get:
Either x = -5 or x = 5.
You can learn more about the difference of two squares method here, and see more solved examples here.

2 comments:

  1. You give two examples that have no real answer.
    It would've been nice if you solved at least one of them.
    x^2 = -81 solved is x = +/- 9i, correct?
    x^2 + 4 = 0 solved is x = +/- 2i, correct?

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    Replies
    1. Correct! Both of the examples (along with other ones in the post) are of pure quadratic expressions. The post explains what are pure quadratic equations. Although the equation `x^2 = -81` does not have any real solutions, its still a pure quadratic equation.

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