I am trying to define the different appraoches to solving quadratic equations. My book says using quadratic formula, completing the sqaure and factoring.

I thought completing the square would be by facotring? How are these two different?

Completing the square is a variation, I suppose, of factoring. The idea is to make the number, which isn't factorable (is that a word?) into one that can be factored. For example,

X^2 + 4x + 4 can be factored into (x+2)^2. So if we had an equation, such as
X^2 + 4x + 3, we rearrange the equation to
X^2 + 4x + ..... = -3
Then we add 4 to both sides, like this,
X^2 + 4x + 4 = -3 + 4
Now the left side is a perfect square
(X + 2)^2 = +1
and we solve for X.
I hope this answers your question.

Great question! Completing the square and factoring are indeed two different methods to solve quadratic equations. Let me explain each approach to you:

1. Factoring: Factoring is the process of expressing a quadratic equation into a product of two binomials, such as (x + a)(x + b). To solve a quadratic equation by factoring, you set the equation equal to zero, try to factor the quadratic expression on the left side into two binomials if possible, and then set each binomial equal to zero. From there, you can solve for the values of x that satisfy the equation.

For example, let's say we have the equation x^2 + 4x + 4 = 0. We can factor this equation as (x + 2)(x + 2) = 0. Setting each binomial equal to zero, we get x + 2 = 0. Solving for x gives us x = -2.

2. Completing the square: Completing the square is a method used to rewrite a quadratic equation in the form of (x + a)^2 = b, where a and b are constants. To solve a quadratic equation by completing the square, you manipulate the equation to create a perfect square trinomial on the left side and then solve for x.

For example, let's say we have the equation x^2 + 6x + 5 = 0. We can complete the square by adding (6/2)^2 = 9 to both sides of the equation: x^2 + 6x + 9 + 5 = 9. Simplifying this gives us (x + 3)^2 = 4. From there, we can take the square root of both sides and solve for x, giving us x + 3 = ±2. Solving for x gives us x = -3 ± 2, which gives us two solutions: x = -1 and x = -5.

In summary, factoring and completing the square are two distinct methods to solve quadratic equations. While factoring involves writing the equation as a product of binomials, completing the square involves rewriting the equation as a perfect square trinomial.