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Roots is just another word for solutions ! e.g. Find the roots of the equation Completing the square: Solution: There are no factors, so we can either complete the square or use the quadratic formula. Using the formula: (1) (2)

The answers from the quadratic formula can be simplified: We have However, 4 is a perfect square so can be square-rooted, so So, 2 is a common factor of the numerator, so Numbers such as are called surds We have simplified the surd

The part is called the discriminant The roots are real and equal. The roots are not real. The roots are real and distinct. ( different ) The Discriminant The formula for solving the quadratic equation is If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

The roots are real and distinct. ( different ) The discriminant e.g. For

The discriminant e.g. For The roots are real and equal.

There are no real roots as the function is never equal to zero. The discriminant e.g. For