Click on A, B, C, or D for more information.

	Description of Discriminant Values	Example	Nature & number of solution(s) or root(s)
Interactive popup. Assistance may be required. A The square root of a perfect square is an integer and therefore rational. When two rational numbers are combined, the result is another rational number. The quotient of two rational numbers is a rational number. If the value of the discriminant is 25, then . If b = 2, then will be -2 +/- 5 = 3 or -7.	Positive & perfect square	4, 9, 25, 36, 49, 64, 10...	Two real, rational roots
Interactive popup. Assistance may be required. B The square root of a positive number that is not a perfect square is an irrational number. When an irrational number is added to or subtracted from a rational number (b), the result is another irrational number. Example: If the value of the discriminant is 24, then , an irrational number. If b = 2, a = 1, and c = -5, then = 2.898979..or -6.898979... Two irrational solutions.	Positive & not perfect square	3, 5, 6, 7, 8, 10…any number not a perfect square	Two real, irrational roots
Interactive popup. Assistance may be required. C The square root of zero is zero, so when either added to or subtracted from -b, the result will be the same. Since a, b and c are real and rational, the one solution will be rational. Example: a = 2, b = 4, c = 2: discriminant = 42 – 4(2)(2) = 16 – 16 = 0 One real, rational solution.	Zero	Zero only	One real, rational root
Interactive popup. Assistance may be required. D When the discriminant is negative, it is necessary to take the square root of a negative number, which results in an imaginary number. When an imaginary number is added to or subtracted from a real number, the result will always be complex (imaginary). Example: a = 2, b = 2, c = 4 discriminant = 22 – 4(2)(4) = 4 – 32 = -28 A negative number under the square root is an imaginary number, and when combined with a real number, yields complex number solutions (For more help, see Module 5 Lesson 7).	Negative	Any negative number: -6, -Pi	Two complex roots

In the quadratic formula, x = -b plus/minus radic b sup 2 minus 4ac over 2a -b ± √b² − 4ac 2a , the square root of the discriminant will be added to and substracted from -b, then divided by 2a.

Click on each of the letters above (A, B, C, and D) to reveal the answer to "What makes these conditions true?"