# How do you solve 10x^2-11x-6=0?

Aug 12, 2016

There are two solutions:
$x = 1.5$ and $x = - 0.40$

#### Explanation:

Since this question is given in standard form, meaning that it follows the form: $a {x}^{2} + b x + c = 0$, we can use the quadratic formula to solve for x:

I think it's worthwhile to mention that $a$ is the number that has the ${x}^{2}$ term associated with it. Thus, it would be $10 {x}^{2}$ for this question.$b$ is the number that has the $x$ variable associated with it and it would be $- 11 x$, and $c$ is a number by itself and in this case it is -6.

Now, we just plug our values into the equation like this:

$x = \frac{- \left(- 11\right) \pm \sqrt{{\left(- 11\right)}^{2} - 4 \left(10\right) \left(- 6\right)}}{2 \left(10\right)}$

$x = \frac{11 \pm \sqrt{121 + 240}}{20}$

$x = \frac{11 \pm 19}{20}$

For these type of problems, you will obtain two solutions because of the $\pm$ part. So what you want to do is add 11 and 19 together and divide that by 20:

$x = \frac{11 + 19}{20}$
$x = \frac{30}{20} = 1.5$

Now, we subtract 19 from 11 and divide by 20:

$x = \frac{11 - 19}{20}$
$x = - \frac{8}{20} = - 0.40$

Next, plug each value of x into the equation separately to see if your values give you 0. This will let you know if you performed the calculations correctly or not

Let's try the first value of $x$ and see if we obtain 0:

$10 {\left(1.5\right)}^{2} - 11 \left(1.5\right) - 6 = 0$

$22.5 - 16.5 - 6 = 0$

$0 = 0$

BOOM, this value of x is correct since we got 0!

Now, let's see if the second value of $x$ is correct:

$10 {\left(- 0.40\right)}^{2} - 11 \left(- 0.40\right) - 6 = 0$

$1.6 + 4.4 - 6 = 0$

$0 = 0$
That value of x is correct as well!

Thus, the two possible solutions are:

$x = - 0.40$
$x = 1.5$

Aug 12, 2016

$\left(5 x + 2\right) \left(2 x - 3\right) = 0$

If (5x + 2) = 0 , then $x = - \frac{2}{5}$

If (2x - 3) = 0, then $x = \frac{3}{2}$

#### Explanation:

Factor the equation by pieces ( Easier than the quadratic if it works)

10 can be factored into 5 x 2 or 10 x 1

6 can be factored into 3 x 2 or 6 x 1

The sum of the factors after being multiplied must add to -11

The bigger factor combination must be negative so 5 x -3 = -15

The smaller factor combination must be positive so 2 x + 2 = +4

• 15 + ( + 2) = -11

• 2 x (-3) = -6

$\left(5 x + 2\right) \left(2 x - 3\right) = 0$

Now that we have the factors we can solve the equation by making each factor equal to 0.

$5 x + 2 = 0 \Rightarrow x = - \frac{2}{5}$
$2 x - 3 = 0 \Rightarrow x = \frac{3}{2}$