# Solving quadratic equations by completing the square

Quadratic equations are also polynomial expressions which are called such because the variable is squared. It has the general formula $ax^2 + bx + c = 0$, but more often than not, it comes in many forms where the terms are rearranged and grouped. You can either just stick to what form you see, or rearrange it so it looks like the general formula, then you can start factoring them.

In this chapter we would be focusing a lot on solving quadratic equations, which can either be through factoring, completing the square or using the quadratic formula. In the first part of the chapter we would be solving quadratic equations through factoring. We have spent a considerable amount of discussing about factoring in previous chapter.

In the second part of the chapter, we will be solving quadratic equations by completing the square. In this method, we need to convert the quadratic equation we have from general form $(ax^2 + bx + c = 0)$ to vertex form $y = a(x - h)^2 + k$ in order to proceed.

For the third part of the chapter we will dwell on the third method which is using the quadratic formula. This method just involves substituting the values in the quadratic formula, $x = \frac{[-b\pm\sqrt{(b^2-4ac)}]}{2a}$. The $b^2-4ac$ is referred to as the discriminant, which would tell us the nature of the roots. The roots can be rational, irrational or imaginary. This is simply a plug and play method where you use the general form, $ax^2 + bx + c = 0$ to get the values you need to substitute in the formula.

The last part of the chapter will be all about solving quadratic inequalities. The trick here is solving them like how we solve equalities. We put the equal sign in place of the inequality sign. The solution or the boundary points that we would get from them would serve as intervals that we would sample to see if they fit into the solution.

### Solving quadratic equations by completing the square

#### Lessons

###### Notes:

###### 4-step approach:

1. isolate X’s on one side of the equation

2. factor out the __leading coefficient__ of $X^2$

3. “completing the square”

• X-side: inside the bracket, add (half of the coefficient of $X)^2$

• Y-side: add [ __leading coefficient__ $\cdot$ (half of the coefficient of $X)^2$ ]

4. clean up

• X-side: convert to perfect-square form

• Y-side: clean up the algebra

__leading coefficient__

__leading coefficient__