Solving a Quadratic in the Form
Some quadratic equations are difficult to solve, and some quadratic equations are easy to solve. Depending on how the equation is provided, certain methods can be very straightforward. Consider the equation . We can find solutions to x by performing inverse operations, just like we solve other multi-step equations:
Unfortunately, not all quadratic equations come in this form. However, we can manipulate the equation to write it as such. This requires a process known as completing the square.
FOIL & Completing the Square
To understand the mechanics of completing the square, it is helpful to connect it to the FOIL process. Let's take the general expression and FOIL it:
The coefficient of the x-term is 2a, and the constant term is a^{2}. If we can manipulate an expression to be in such a form, then we can write it equivalently as (x+a)^{2}. This is the goal of completing the square.
Solving a Quadratic by Completing the Square
The process of completing the square follows a specific set of steps in order to convert the equation into one similar to our very first example:
Let's apply these steps to the equation:
The last step is a bit tricky. We were able to write the right side of the equation as a binomial squared, because of the relationship between the coefficients and the constant term. When is doubled, it is equivalent to the x-term coefficient in expanded form, , AND when it is squared, it equals to constant term in the expanded form,
Now that we have converted an equation from standard form into a binomial squared, we can solve this equation following the same procedure as our very first example:
the process of converting a quadratic equation in the form ax^2+bx+c =0 into an expression involving a perfect square trinomial