Final answer:
Creating a perfect square trinomial involves squaring a binomial, resulting in an expression of the form x^2 + 2ax + a^2. Squaring the terms separately and combining them yields the perfect square. The process adapts for exponents by squaring the digit term as usual and multiplying the exponent by 2.
Step-by-step explanation:
To create a perfect square trinomial, you start with a binomial of the form (x + a)^2 or (x - a)^2 and apply the squaring process. A perfect square trinomial is an expression obtained when you square a binomial, resulting in the form x^2 + 2ax + a^2. Let's look at the process of squaring the binomial (x + a):
- First, square the first term: (x)^2 = x^2.
- Second, multiply the two different terms together and then double the result: 2(x)(a) = 2ax.
- Finally, square the last term: (a)^2 = a^2.
Combining these steps gives you the perfect square trinomial: x^2 + 2ax + a^2.
To illustrate this with numbers, let's take (x + 3)^2:
- Square x: x^2.
- Multiply x by 3 and then double: 2(3)x = 6x.
- Square 3: 3^2 = 9.
So, the perfect square trinomial becomes x^2 + 6x + 9.
In cases where we're dealing with exponents, the rule is similar but adapted to exponential terms. For example, when squaring a term like 3^2, you square the numeric part as usual and then multiply the exponent by 2: (3^2)^2 = 3^(2*2) = 3^4. For more complex quadratic equations, a perfect square trinomial can also be created by 'completing the square,' which often makes it easier to solve the equation.