To find the number of real solutions to a given equation, one must first determine what type of equation it is. If it is a linear equation, then there will be exactly one real solution. If it is a quadratic equation, then there can be either two or zero real solutions. Finally, if the equation is cubic, then there can be either three, two, or zero real solutions. In order to determine how many real solutions there are for equations of each type, one must use the following methods:
For linear equations:
There will be exactly one real solution if the discriminant (b^2-4ac) is positive. There will be no real solutions if the discriminant is negative.
For quadratic equations:
There will be two real solutions if the discriminant (b^2-4ac) is positive. There will be no real solutions if the discriminant is negative and at least one root is imaginary. There will be only one repeated root if the discriminant is zero and both roots are imaginary.
For cubic equations:
If all 3 roots are real and unequal (R), then there are 3real roots(3R). If 2roots are equal(RR),then 1 R and 1 I with 2 complex conjugate roots(1R+2I).If all 3rootsare equal(RRR),then all 3rootsarerealandequal(3R).