The Discriminant of the Quadratic Equation | Quadratic Equation in One Variable | Precalculus

The Discriminant of the Quadratic Equation For ax^2+bx+c=0, where a, b, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula:  b^2-4ac.  It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect. The following table relates the value of the discriminant to the nature of the roots of a quadratic equation. Value of Discriminant Nature of the Roots of the Quadratic Equation b^2-4ac=0 roots are real and equal; a double solution b^2-4ac(greater than)0 roots are real and unequal b^2-4ac(less than)0 roots are imaginary and unequal; they are complex conjugates of each other. Example 6: Determine the character of the roots of each of the following equations. a) 3x^2-2x-6=0 b) 4x^2-12x+9=0 c) 2x^2+6x+7=0 Solution: a) For the given equation, a=3,b=-2 and c=-6. Thus b^2-4ac = (-2)^2-4(3)(-6) = 76 The discriminant is positive. Hence, the roots are real and unequal. b) The discriminant is b^2-4ac = (-12)^2-4(4)(9) = 0 The discriminant is zero; therefore, the roots are equal real numbers. c) The discriminant is b^2-4ac = (6)^2-4(2)(7) = -20 The discriminant is negative; therefore, the roots are imaginary and unequal; they are complex conjugates of each other.