How do you solve #-3n^2 + 5n - 2= 0# using the formula?
First we need to remember the formula:
In this problem we have
("Minus b, plus or minus the square root of b squared minus 4 a c all over 2a".)
In order to use the formula, we need to identify
You may notice that every number I substituted is in parentheses in this first step. I think that is a good habit to develop for all formulas.
Now we have some simplification/arithmetic to do (I'll remove the colors now)
# = (-5 +- sqrt (25-(-12)(-2)))/(-6)#
# = (-5 +- sqrt (25-(24)))/(-6)#
# = (-5 +- sqrt 1)/(-6)#
# = (-5 +- 1)/(-6)#.
There are two solutions:
And the other solution is
At the point where we had:
We could have made the denominator positive, be doing this:
Remember that writing
The opposite of plus or minus 1, is minus or plus 1. Which is surely the same as plus or minus 1.
Finally, notice that when we finish the arithmetic, we get the same answers:
(We got the same numbers in the opposite order.)
Solve -3x^2 + 5x - 2 = 0
For this type of equation, we don't need a lengthy solving process.
Use the shortcut.
When (a + b + c = 0), one real root is (1) and the other is (c/a = 2/3).
Remind of Shortcut Rule.
- When (a + b + c = 0): 2 real roots -> (1) and (c/a)
#7x^2 - 15x + 8 = 0.#
(a + b + c = 0)--> 2 real roots: (1) and
#(c/a = 8/7)#
- When (a - b + c = 0) --> 2 real roots (-1) and (-c/a).
#19x^2 + 8x - 11 = 0#
(a - b + c = 0) -> 2 real roots: (-1) and
#(-c/a = 11/19)#
The shortcut will save us a lot of work and effort.