18.3 Solving quadratic equations using the quadratic formula

Quadratic equations are also polynomial expressions which are called such because the variable is squared. It has the general formula ax2+bx+c=0ax^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 (ax2+bx+c=0)(ax^2 + bx + c = 0) to vertex form y=a(xh)2+ky = 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=[b±(b24ac)]2ax = \frac{[-b\pm\sqrt{(b^2-4ac)}]}{2a}. The b24acb^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, ax2+bx+c=0ax^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 using the quadratic formula

Not sure if you should solve the quadratic equation by factoring or completing the square? No worries. You can always use the quadratic formula. The beauty of the quadratic formula is that it can always give you the answer no matter if the quadratic equations can be factored or not.


Quadratic Formula
For the quadratic equation: ax2+bx+c=0a{x^2} + bx + c = 0
the solutions are: x=b±b24ac2ax = \frac{{ - b\; \pm \;\sqrt {{b^2} - 4ac} }}{{2a}}
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Solving quadratic equations using the quadratic formula

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