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