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

Jul 1, 2015

First we need to remember the formula:

Explanation:

For $a {x}^{2} + b x + c = 0$, the solutions are given by:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

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 $a , b , \text{and } c$

In: $- 3 {n}^{2} + 5 n - 2 = 0$, we have:

$\textcolor{red}{a = - 3}$ and $\textcolor{b l u e}{b = 5}$ and $\textcolor{g r e e n}{c = - 2}$, 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)

$= \frac{- 5 \pm \sqrt{25 - \left(- 12\right) \left(- 2\right)}}{- 6}$

$= \frac{- 5 \pm \sqrt{25 - \left(24\right)}}{- 6}$

$= \frac{- 5 \pm \sqrt{1}}{- 6}$

$= \frac{- 5 \pm 1}{- 6}$.

There are two solutions:

One is $\frac{- 5 + 1}{-} 6$ which simplifies to $\frac{- 4}{-} 6 = \frac{2}{3}$

And the other solution is $\frac{- 5 - 1}{-} 6$ which simplifies to $\frac{- 6}{-} 6 = 1$

note
At the point where we had:
$x = \frac{- 5 \pm 1}{- 6}$.
We could have made the denominator positive, be doing this:

$\frac{- 5 \pm 1}{- 6} = \frac{\left(- 1\right) \left(- 5 \pm 1\right)}{6}$

Now $- \left(- 5\right) = 5$, but what about $- \left(\pm 1\right)$?

Remember that writing $\pm 1$ 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

$\frac{- 5 \pm 1}{- 6} = \frac{\left(- 1\right) \left(- 5 \pm 1\right)}{6} = \frac{5 \pm 1}{6}$.

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

$\frac{5 + 1}{6} = \frac{6}{6} = 1$ and $\frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}$

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

Jul 2, 2015

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 : $7 {x}^{2} - 15 x + 8 = 0.$
(a + b + c = 0)--> 2 real roots: (1) and $\left(\frac{c}{a} = \frac{8}{7}\right)$
2. When (a - b + c = 0) --> 2 real roots (-1) and (-c/a).
Example: $19 {x}^{2} + 8 x - 11 = 0$
(a - b + c = 0) -> 2 real roots: (-1) and $\left(- \frac{c}{a} = \frac{11}{19}\right)$

The shortcut will save us a lot of work and effort.