How many distinct real roots can there be for a quadratic equation of the form ax^2 + bx + c = 0?

Question

 How many distinct real roots can there be for a quadratic equation of the form ax^2 + bx + c = 0? 

 

I need the answer with examples so I can understand.

Answer 1

There are as many roots as the highest exponent in the equation.

A quadratic equation of the form ax²+bx+c can have zero, one, or two distinct real roots;

 

This is because the roots can be either real, complex or one root with multiplicity two.

 

Example:

 

x²-8x+16 = 0

x²-8x+16 = (x-4)(x-4) = (x-4)²

There is one real root, x=4, with a multiplicity of 2

 

Example:  x²-4x-21= (x+3)(x-7) = 0

Two real roots:  x=-3, x=7

 

Example:  x²+3x+3 = 0

x = [-3 ±√((3)²-4(1)(3))]/2(1) = [-3±√(-3)]/2 

= (-3 ±i√3)/2

 

In general the basic rule is:

 

For quadratic equation ax²+bx+c=0

x= (-b ±√(b²-4ac))/2a

 

If the discriminant - part under square root - is negative then

there will be complex roots 

 

If the discriminant is positive there are two real roots

 

If the discriminant equals zero there is one real root of multiplicity two

 

 

Answer 2

There are either 0,1, or 2 real roots. This number is determined by the determinate which is 

 

b²-4ac. 

 

If it is >0 then there are 2 real roots,

 

If it is =0 then there is 1 real roots,

 

If it is <0 then there are 0 real roots,

 

you can use the quadratic formula to come up w examples.