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3. Solving Quadratic Equations

Solving quadratic equations by completing the square

In previous chapters, we had discussed Quadratic Equations, starting with understanding their characteristics, knowing how their graph looks like, familiarizing ourselves with the general form and learning how the different factoring methods work.

Given that we have already established those concepts, we will now learn how to solve quadratic equations. We could always resort to the easiest way of solving them by using a free quadratic formula calculator, but it still pays to know how to solve them manually.

In this chapter we will learn the three ways to solve a quadratic equation. The first method is through factoring given that the quadratic that we have is factorable, otherwise, the method is not applicable.

We spent some time on discussing about factoring quadratics in previous chapters, we it's expected that you already understand this method. If the quadratic is in its general form, which is $ax^2 + bx + c = 0$, then we simply need to factor the quadratic, and solve for the variable given like in the case of $x^2 + 5x + 6$, we know that the factors are (x+2) and (x+3) which would lead us to the value of x, which are 2, and 3. If it isn’t in its general form then we need to rearrange it to make things simpler.

The second method is completing squares, which we had discussed in previous chapter. This method requires us 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.

The last method we will learn about is using 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.

Solving quadratic equations by completing the square

When a quadratic equation cannot be factorized, we can use the method of completing the square to solve the equation.