Algebra Know-It-ALL

We’re not quite done yet, because this formula can be simplified. We have a common
denominator, 2a, in the difference of the two ratios on the right side. We can therefore rewrite
the above formula as

x= [±(b^2 − 4 ac)1/2−b] / (2a)

In most texts, the numerator is written with −b first, like this:

x= [−b± (b^2 − 4 ac)1/2] / (2a)

An example

Now it’s time to solve a specific equation using the quadratic formula. Let’s try this:

9 x^2 + 12 x− 21 = 0

In this case, we have a= 9, b= 12, and c=−21. We can plug these numbers into the quadratic
formula and grind it out:

x= [−b± (b^2 − 4 ac)1/2] / (2a)

= {− 12 ± [12^2 − 4 × 9 × (−21)]1/2} / (2 × 9)

= {− 12 ± [144 − (−756)]1/2} / 18

= (− 12 ± 900 1/2) / 18

= (− 12 ± 30) / 18

= (− 12 + 30) / 18 or (− 12 − 30) / 18

= 18/18 or −42/18

= 1 or −7/3

The solution set is therefore {1,−7/3}. You can check these roots by plugging them back into
the original equation.

Another example

Now let’s see what happens when we solve this equation with the quadratic formula:

4 x^2 − 24 x+ 36 = 0

Here, we have a= 4, b=−24, and c= 36. Plugging in and grinding out, we obtain

x= [−b± (b^2 − 4 ac)1/2] / (2a)

= {−(−24)± [(−24)^2 − 4 × 4 × 36]1/2} / (2 × 4)

= [24 ± (576 − 576)1/2}/ 8

= (24 ± 0 1/2}/ 8

= 24/8

= 3

The solution set is {3}. There is only one root. Feel free to check it!

The Quadratic Formula 377