How to Solve a Perfect Square Binomial
Solving a perfect square binomial is a fundamental skill in algebra that can be easily mastered with a clear understanding of the concept. A perfect square binomial is a quadratic expression that can be expressed as the square of a binomial. This article will guide you through the steps to solve a perfect square binomial effectively.
Firstly, it is essential to recognize a perfect square binomial. A perfect square binomial is in the form of (a + b)^2 or (a – b)^2, where ‘a’ and ‘b’ are real numbers. When expanded, these expressions result in the quadratic equation a^2 + 2ab + b^2 or a^2 – 2ab + b^2, respectively. The key feature of a perfect square binomial is that the middle term is always twice the product of the first and last terms.
To solve a perfect square binomial, follow these steps:
1. Identify the binomial and determine whether it is in the form (a + b)^2 or (a – b)^2.
2. Expand the binomial using the formula (a + b)^2 = a^2 + 2ab + b^2 or (a – b)^2 = a^2 – 2ab + b^2.
3. Simplify the expanded expression by combining like terms.
4. If the perfect square binomial is in the form (a + b)^2 or (a – b)^2, the solution will be a = a and b = b, since the binomial is already a perfect square.
5. If the perfect square binomial is in the form ax^2 + bx + c, factor the expression into the form (a + b)^2 or (a – b)^2.
6. Solve for ‘a’ and ‘b’ by equating the coefficients of the like terms.
Let’s illustrate these steps with an example:
Example: Solve the perfect square binomial 4x^2 – 12x + 9.
1. Identify the binomial: The binomial is 4x^2 – 12x + 9.
2. Expand the binomial: (2x)^2 – 2 2x 3 + 3^2 = 4x^2 – 12x + 9.
3. Simplify the expanded expression: The expression is already simplified.
4. Recognize the perfect square binomial: The binomial is in the form (2x – 3)^2.
5. Factor the expression: 4x^2 – 12x + 9 = (2x – 3)^2.
6. Solve for ‘a’ and ‘b’: a = 2x, b = 3.
In conclusion, solving a perfect square binomial involves recognizing the binomial’s form, expanding it, and then simplifying the expression. By following these steps, you can effectively solve any perfect square binomial.
