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## Using the Discriminant to predict the roots of a quadratic equation

The discriminant of a quadratic equation is the value under the square root sign in the quadratic formula.

Remember the quadratic formula for an equation in the form ax + bx + c = 0 is: From this formula the discriminant is: b - 4ac

When you evaluate the discriminant for a quadratic equation, if the result is:

 positive You will have 2 different real solutions to the equation If this number is a perfect square number, there will be 2 different rational answers. If this number is a not perfect square number, there will be 2 different irrational answers. zero You will have 1 real, rational solution to the equation - that is, there will be a repeated answer negative You will have no real solutions to the equation (only imaginary answers)

Examples:

Use the discriminant to predict the roots of the following equations:

 1. x + 7x + 12 = 0 a = 1 b = 7 c = 12 b - 4ac = 7 - 4(1)(12) = 49 - 48 = 1

Since the result is positive, there should be 2 different real solutions.

In fact, there will be 2 different rational solutions because 1 is a perfect square number.

(Perfect square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc)

 2. x + 7x + 3 = 0 a = 1 b = 7 c = 3 b - 4ac = 7 - 4(1)(3) = 49 - 12 = 37

Since the result is positive, there should be 2 different real solutions.

In fact, there will be 2 different irrational solutions because 37 is not a perfect square number.

 3. x + 4x + 4 = 0 a = 1 b = 4 c = 4 b - 4ac = 4 - 4(1)(4) = 16 - 16 = 0

Since the result is zero, there should be only one real, rational solution

 4. x - x + 4 = 0 a = 1 b = -1 c = 4 b - 4ac = (-1) - 4(1)(4) = 1 - 16 = -15

Since the result is negative, there should be no real solutions.