Need help with these 2 "equations" questions (alg. 2 stuff)?

Question

How can you tell if you'll get imaginary or complex numbers for a solution to the quadratic equation?

How can you deal with an equation involving rational expressions("fractions" with polynomials)?

Answer 1

(1) Let the equation be ax^2 + bx + c = 0

Then its solution is given by the quadratic formula

x = [- b +/- sqrt(b^2 - 4ac)]/(2a)

Since the square root of a negative quantity is an imaginary quantity, the type of solution you will get depends on the value of b^2 - 4ac.

If b^2 - 4ac is negative then the solution contains a square root of a negative number and is therefore complex. (A complex number is the sum of a real number and an imaginary number.)

If b^2 - 4ac is positive or zero then the solution is real as it contains a square root of a positive number or zero.

(2) Do you mean, how to get rid of the denominators? By multiplying throughout by any common multiple of the denominators. For example if the equation is

(1/3)x^2 + (1/2)x + 1 = 0

then the denominators are 3 and 2, so to get rid of them we would multiply throughout by 3 x 2 = 6. We get:

(6 x 1/3)x^2 + (6 x 1/2)x + (6 x 1) = 6 x 0

i.e.

2x^2 + 3x + 6 = 0