Question

Determine the type and number or roots for the following. 3x^2+x-1=0

Answer 1

For:
ax^2+bx+c=0

Calculate the discriminant of the quadtratic equation:

b^2-4ac

Then check it's value:
if it's < 0 --> there will be no real roots
if it's = 0 --> there will be one real root
if it's > 0 --> there will be two real roots
if it's a perfect square --> two real roots, which are rational numbers

Answer 2

IS this the formula or is this the answer? I have never done this before so I am not fimiliar with it.

Answer 3

OK

In general, you have this representation of a quadratic:
ax^2+bx+c=0

In your case it's:
3x^2+x-1=0

So:
a=3
b=1
c=-1

Now, calculate the discriminant value using the values above for a,b,c:

b^2-4ac

what do you get ?

Answer 4

?

Answer 5

you are told that
a=3 , b=1, c=-1

How much is:
b^2-4ac=?

Do you know how to do this ?

Answer 6

13

Answer 7

correct !

Answer 8

now what do I do?

Answer 9

Check the discriminant value:
if it's < 0 --> there will be no real roots
if it's = 0 --> there will be one real root
if it's > 0 --> there will be two real roots
if it's a perfect square --> two real roots, which are rational numbers

you know the discriminant value is 13.
so look at the above and you will know the information about the roots which the question is asking about.

Answer 10

So this is saying that there will be no real roots?

Answer 11

well, 13 is greater than 0 - right ?
so look at the one that says:
if it's > 0 --> ....

Answer 12

so there will be two real roots

Answer 13

yes that's right.
now is 13 a perfect square or not ?

Answer 14

no because it is prime

Answer 15

good - so you also know that the roots are not going to be rational numbers.

Answer 16

right

Answer 17

now here's the full quadratic equation:

First, we have the discriminant,
which you already calculated, and is 13 in our example.
\[\Delta=b^2-4ac\]

Now the roots will be:
\[x _{1,2}=\frac{-b \pm \sqrt{\Delta}}{2a}\]

Answer 18

What is the x1,2

Answer 19

now you can see that the discriminant is placed inside a square root.

All these properties of the discriminant are a result of this.

if it's < 0 --> there will be no real roots
if it's = 0 --> there will be one real root
if it's > 0 --> there will be two real roots
if it's a perfect square --> two real roots, which are rational numbers