How do you do the quadratic formula?

Question

I just need to know how to use the quadratic formula so I can begin to understand it in a better and easier way.

Answer 1

The quadratic formula is used when a quadratic equation (ax²+bx+c = 0) is un-factorable.

 

If you were given, for example, the equation 2x² - 5x - 3 = 0, you wouldn't need the quadratic formula. You could just factor the equation to (2x + 1) (x - 3)=0 and solve from there (by setting each binomial equal to zero and solving for your two solutions).

 

Occasionally, however, you may be given a problem like 2x² - 3x - 1 = 0 which would be considered un-factorable. In this case, you would use the quadratic formula

 

 X = (-b ⁺_- sqrt b²-4ac)/2a

 

matching parts of

     ax²+bx+c=0 and

     2x² -3x -1=0

we find a = 2, b = -3, and c = -1

 

Now, just substitute those values into the quadratic formula and you have your answer(s)

 

    x =[ -(-3) ⁺_- sqrt of (-3)² -4 (2)(-1)]/2(2)

 

    x = [3 ⁺_- sqrt of 17]/4

 

Technically, you could always use the quadratic formula (even on equations that are factorable...it would just be A LOT more work than necessary)

Answer 2

To help you out, let's use an already factored equation to see what our final answers are going to be, then work backwards into FOIL, and from there the quadratic formula to see that we end up with the exact same numbers as in the beginning:

(x + 3)(x - 2) = 0

(This means our final answers should be x = -3 & 2)

FOILing that, we get: x² - 2x + 3x - 6

Simplified:

x² + x - 6

 

Here's what the quadratic formula looks like:

x = (-b ± √(b² - 4ac)) / 2a

a = 1 (the implied value in front of x²)

b = 1 (in front of the x, we have a plus sign and an implied 1 again)

c = -6 (our constant on the end)

 

With those values, we put them into the quadratic formula:

x = -1 ± √(1² - 4 * 1 * -6) / (2 * 1)

 

Now the Q-F may look scary with all of this information, but all we need to do is take small steps at a time, and it will clean up right quick. Let's simplify some of these parts as part of the cleaning up:

1² = 1

4 * 1 * -6 = -24

2 * 1 = 2

That gives us so far:

x = (-1 ± √(1 - (-24))) / 2

 

Now let's focus on simplifying the square root:

x = (-1 ± √(1 + 24)) / 2

x = (-1 ± √25) / 2

x = (-1 ± 5) / 2

 

That leaves us with:

x = (-1 + 5) / 2 and x = (-1 - 5) / 2

Now to focus on the adding of the root:

x = 4 / 2

x = 2

And now we look at the subtracting of the root:

x = (-1 - 5) / 2

x = -6 / 2

x = -3

 

If we look back to our original solutions, they are indeed supposed to be 2 and -3.

You may be asking yourself "Why do I need to know this? Can't I just factor out?" While that can indeed be useful, the a-variable can be a multitude of numbers, and the higher and less prime it is, the more possible answers you're going to have to go through to find the actual solution. Additionally, not all quadratics factor cleanly. You can end up with an irrational number like √17 as your work your way through the quadratic formula, and that can happen. That's why it can be a useful tool when factoring out can be a grueling process.