How exactly do you find/solve a Perfect Square Trinomials and factoring them?
To find and solve a perfect square trinomial, you'll need to understand what a perfect square trinomial is and how to factor it.
A perfect square trinomial is a quadratic expression of the form ax^2 + bx + c, where the first and last term are perfect squares, and the middle term is twice the product of the square root of the first term and the square root of the last term.
The general form of a perfect square trinomial is (a√x ± b)^2, where a, b, and x are integers or variables.
To factor a perfect square trinomial, you follow this process:
1. Identify if the trinomial is indeed a perfect square trinomial by comparing it to the general form.
2. Take the square root of the first and last term, which will give you two square root expressions.
3. Write these square roots expressions as (a√x ± b), where a and b are integers or variables.
4. Multiply the two expressions from step 3 to obtain a quadratic expression.
Let's walk through an example:
Example: Solve the perfect square trinomial x^2 + 6x + 9.
1. Compare the trinomial with the general form, (a√x ± b)^2. You'll notice that the first term (x^2) is a perfect square (x^2 = (x)^2) and the last term (9) is also a perfect square (9 = (3)^2). Additionally, the middle term (6x) can be expressed as 2 times the product of the square roots: 2(x)(3) = 6x. Therefore, this trinomial is a perfect square trinomial.
2. Take the square root of the first term x^2 and the last term 9. The square root of x^2 is x, and the square root of 9 is 3.
3. Write the square root expressions as (a√x ± b). Therefore, we have (x + 3)^2.
4. Multiply the two expressions to obtain (x + 3)(x + 3) or (x + 3)^2, which is our final answer.
In summary, to find and solve a perfect square trinomial, compare the quadratic expression to the general form of a perfect square trinomial, determine if it meets the criteria, take the square root of the first and last term, express them as (a√x ± b), and multiply these expressions to obtain the factored form of the trinomial.