How to Complete Perfect Square Trinomial
In mathematics, a perfect square trinomial is a polynomial of the form \(a^2 + 2ab + b^2\), where \(a\) and \(b\) are real numbers. This expression is also known as a square of a binomial. Completing the square is a fundamental technique used to solve quadratic equations and factorize polynomials. In this article, we will discuss how to complete a perfect square trinomial and its applications.
Understanding the Concept
To complete a perfect square trinomial, we need to identify the values of \(a\) and \(b\) in the expression \(a^2 + 2ab + b^2\). The first step is to ensure that the quadratic term \(a^2\) is a perfect square. If it is not, we can multiply the entire expression by a constant to make it a perfect square.
For example, consider the expression \(3x^2 + 6x + 2\). To make the quadratic term a perfect square, we can multiply the entire expression by \(\frac{1}{3}\):
\[\frac{1}{3}(3x^2 + 6x + 2) = x^2 + 2x + \frac{2}{3}\]
Now, we have a perfect square trinomial in the form \(x^2 + 2x + \frac{2}{3}\).
Identifying the Values of \(a\) and \(b
Once we have a perfect square trinomial, we can identify the values of \(a\) and \(b\) by comparing the given expression with the standard form \(a^2 + 2ab + b^2\). In our example, \(a = x\) and \(b = 1\).
Completing the Square
To complete the square, we need to add and subtract the square of \(b\) (i.e., \(b^2\)) to the expression. This will create a perfect square trinomial on one side of the equation:
\[x^2 + 2x + 1 – 1 + \frac{2}{3} = (x + 1)^2 – \frac{1}{3}\]
Now, we have successfully completed the square for the expression \(3x^2 + 6x + 2\). The resulting expression is \((x + 1)^2 – \frac{1}{3}\).
Applications of Completing the Square
Completing the square is a valuable technique with various applications in mathematics. Some of the main uses include:
1. Solving quadratic equations: By completing the square, we can easily find the roots of a quadratic equation.
2. Factoring polynomials: Completing the square can help us factorize certain types of polynomials.
3. Deriving formulas: The process of completing the square is used to derive important formulas, such as the quadratic formula.
In conclusion, completing the square is a fundamental technique in mathematics that helps us solve quadratic equations and factorize polynomials. By understanding the concept and following the steps outlined in this article, you can effectively complete a perfect square trinomial and apply it to various mathematical problems.
