Explain how to determine the number and type of roots (both real and imaginary) from the graph of a quadratic function.
There are six types of roots.
1) Real and unequal.
When and are real numbers, and the discriminant is positive, then the roots of the quadratic equation are real and unequal.
2) Real and equal.
When and are real numbers, and the discriminant is zero, then the roots of the quadratic equation are real and equal.
3) Unequal and imaginary.
When and are real numbers, and the discriminant is negative, then the roots of the quadratic equation are unequal and not real. In this case, we say that the roots are imaginary.
4) Real, rational and unequal.
When and are real numbers, and the discriminant is positive and perfect square, then the roots of the quadratic equation are real, rational and unequal.
5) Real, irrational and unequal.
When and are real numbers, and the discriminant is positive but not a perfect square then the roots of the quadratic equation are real, irrational and unequal.
Here the roots form a pair of irrational conjugates.
6) Irrational
When and are real numbers, and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation are irrational.
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