Question

How do you solve `-3n^2 + 5n - 2= 0` using the formula?

Answer 1

First we need to remember the formula:

Explanation:

For `ax^2+bx+c=0`, the solutions are given by:

`x=(-b +- sqrt(b^2-4ac))/(2a)`

In this problem we have `n` instead of `x`, but the formula is the same.

("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 `a, b, "and " c`

In: `-3n^2 + 5n - 2= 0`, we have:

`color(red) (a=-3)` and `color(blue)(b=5)` and `color(green)(c=-2)`, so we substitute in the formula:

`n = (-color(blue)((5))+-sqrt(color(blue)((5))^2-4color(red)((-3))color(green)((-2))))/(2color(red)((-3))`

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)

`n = (-(5)+-sqrt((5)^2-4(-3)(-2)))/(2(-3)`

`= (-5 +- sqrt (25-(-12)(-2)))/(-6)`

`= (-5 +- sqrt (25-(24)))/(-6)`

`= (-5 +- sqrt 1)/(-6)`

`= (-5 +- 1)/(-6)`.

There are two solutions:

One is `(-5+1)/-6` which simplifies to `(-4)/-6 = 2/3`

And the other solution is `(-5-1)/-6` which simplifies to `(-6)/-6 = 1`

note
At the point where we had:
`x = (-5 +- 1)/(-6)`.
We could have made the denominator positive, be doing this:

`(-5 +- 1)/(-6) = ((-1)(-5 +- 1))/6`

Now `-(-5) = 5`, but what about `-(+-1)`?

Remember that writing `+-1` is just a short way of writing the wto numbers `+1` and `-1`, so what we get, in words, is:

The opposite of plus or minus 1, is minus or plus 1. Which is surely the same as plus or minus 1.
So

`(-5 +- 1)/(-6) = ((-1)(-5 +- 1))/6 = (5+-1)/6`.

Finally, notice that when we finish the arithmetic, we get the same answers:

`(5+1)/6 = 6/6=1` and `(5-1)/6 = 4/6 = 2/3`

(We got the same numbers in the opposite order.)

Answer 2

Solve -3x^2 + 5x - 2 = 0

Explanation:

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.

1. When (a + b + c = 0): 2 real roots -> (1) and (c/a)
   Example : `7x^2 - 15x + 8 = 0.`
   (a + b + c = 0)--> 2 real roots: (1) and `(c/a = 8/7)`
2. When (a - b + c = 0) --> 2 real roots (-1) and (-c/a).
   Example: `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.