Question

How do I state the number of complex roots, the possible number of real and imaginary roots, and the possible rational roots for each equation? How do I find all roots?

Answer 1

Answer:

$sqt(b^2-4ac)$

Step-by-step explanation:

The expression which is given in the answer is called the discriminant. Because it discriminate all the types of roots which can be derived from any quadratic equation. Now imagine a standard quadratic equation:

$ax^2+bx+c$

The value of x can be derived from quadratic formula which is

x = -b±[(b^2-4ac)^(-1/2)]/2a

Now take out discriminant, which is

$sqrt(b^2-4ac)$

if you put values of a,b and c from a quadratic equation,

you can possibly have three types of discriminant values. Lets represent discriminant as D

So, D can be 0

D can be any real number greater than 0

D can be any real number less than 0

Now we will understand each.

If D = 0, you can see that the both roots will be equal, because in standard quadratic formula, discriminant will make no change either added or subtracted. So we will have equal roots if discriminant is zero. Equal roots means we will only have one value of roots.

If D is any real number greater than zero, it will be simply any real roots. There will be two values of roots, because you will first add up discriminant in quadratic formula and then subtract, because this sign ± means you have two perform both functions one by one.

If D is real number less than 0, we know that by applying square root, it will be converted into an imaginary number, a number with iota, e.g 4i. You will have two imaginary roots here due to this sign ± means you have two perform both addition and subtraction one by one.

Now the complex numbers are basically combination of real and imaginary numbers, so if discriminant has value less than zero, you would have imaginary as well as complex roots and there number will be two because due to this sign ± you have to perform addition and subtraction again.