What are the pros and cons of completing the square as a way to solve quadradic equations?
I find the quadratic equation
x = [-b +/- sqrt(b^2-4ac)]/2a
the easiest to use, unless a way of factoring is obvious. It is derived by completing the square, after all. It tells you the number of real roots right away (from the value of b^2 - 4ac). The hard part is memorizing it, but after a while it becomes routine.
Completing the square is not the easiest way to solve quadratic equations; its strength lies in the fact that the process is repetitive and predictable.
Here is the best news: completing the square ALWAYS (SAY ALWAYS) will work, unlike the factoring method, which of course, requires that the trinomial be factorable.
Thank you.
Cool rule, thank you.
Thank you for the example.
Completing the square is a useful method for solving quadratic equations. Here are the pros and cons of using this method:
Pros:
1. It guarantees that you will find the exact solutions of the quadratic equation.
2. It works for all quadratic equations, whether they are factorable or not.
3. It allows you to manipulate the equation algebraically, making it easier to solve.
Cons:
1. It can be time-consuming and involve multiple steps, particularly if the coefficients of the quadratic equation are large or contain fractions.
2. It requires precision and careful algebraic manipulation, making it prone to human errors.
3. It might not be the most efficient method for solving quadratic equations, especially when there are alternative techniques available, such as factoring or using the quadratic formula.
To complete the square, follow these steps:
1. Ensure that the coefficient of the quadratic term (x^2) is 1. If it is not, divide the entire equation by that coefficient.
2. Move the constant term to the other side of the equation, so the right side is zero.
3. Take half the coefficient of the linear term (x), square it, and add this value to both sides of the equation. This step is called "completing the square."
4. Rewrite the left side of the equation as a perfect square trinomial and simplify the right side.
5. Solve for x by taking the square root of both sides and applying the square root property.
6. Write down the two possible solutions for x.
Remember, completing the square is just one of the methods to solve quadratic equations. Depending on the specific equation and its characteristics, you may choose an alternative approach that better suits the situation.
I use the following rule:
If the coefficient of the squared term is 1 and the coefficient of the first degree term is even, then I would use completing the square, otherwise just use the quadratic formula
e.g.
x^2 - 12x -5 = 0
x^2 - 12x = 5
x^2 - 12x + 36 = -5 + 36
(x-6)^2 = 31
x = 6 ± √31
In this case this method is actually faster and easier than using the formula, since the formula answer has to be broken down to lowest terms.