Generally, a discriminant is a function of the coefficients of a polynomial. It determines the nature of solutions an equation can have without exactly finding them.
In a quadratic formula, the discriminant is only a part of the quadratic formula within the square root. For a quadratic equation ax2 + bx + c = 0, b2 – 4ac is the discriminant (D), as shown in the diagram below.
Discriminant of a Quadratic Equation
Thus, to find the discriminant of a quadratic equation, follow the following steps:
Step 1: Compare the given quadratic equation with its standard form ax2 + bx + c = 0 and find the values of a, b and c
Step 2: Substitute the values in the discriminant b2 – 4ac to get the result
Let us find the discriminant of the quadratic equation x2 + 10x + 16 = 0
Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 1, b = 10, c = 16
Substituting the values in the discriminant b2 – 4ac, we get
= (10)2 – 4 × 1 × 16
= 100 – 64
= 36
Thus, the discriminant of the quadratic equation x2 + 10x + 16 = 0 is 36
What is the discriminant of the quadratic equation 3 – 4x – 6x2 = 0?
Solution:
Rearranging the equation to -6x2 – 4x + 3 = 0 Comparing the given quadratic equation with its standard form ax2 + bx + c = 0 we get, a = -6, b = -4, c = 3 Substituting the values in the discriminant b2 – 4ac, we get = (-4)2 – 4 × -6 × 3 = -56 Thus, the discriminant of the quadratic equation 3 – 4x – 6x2 = 0 is 36
Determine the discriminant for the quadratic equation -3x2 + 4x + 1 = 0
Solution:
Comparing the given quadratic equation with its standard form ax2 + bx + c = 0 we get, a = -3, b = 4, c = 1 Substituting the values in the discriminant b2 – 4ac, we get = (4)2 – 4 × -3 × 1 = 28 Thus, the discriminant of the quadratic equation -3x2 + 4x + 1 = 0is 36
Discriminant also tells the number of solutions a quadratic equation has and the nature of the graph it produces in the coordinate plane. There can be three different situations:
If the Discriminant is Positive
If b2 – 4ac > 0, the quadratic equation has 2 real solutions
It happens because the square of a positive number always gives a real number. So, when the discriminant is greater than 0 it is 2 roots that real and distinct. The graph of such an equation crosses through the x-axis at two points.
If the Discriminant is Negative
If b2 – 4ac < 0, the quadratic equation has no real solutions but two different complex or imaginary roots
It happens so because the square of a negative number gives an imaginary number. For example, ${\sqrt{-9}}$ = 3i. So, when the discriminant is less than 0, it has two distinct and complex roots. The graph of such an equation does not cross the x-axis.
If the Discriminant is Zero
If b2 – 4ac = 0, the quadratic equation has one real root
It happens so because the square of 0 is 0. The graph of such an equation touches the x-axis at one point.
Determine whether each of the following quadratic equations has two real roots, one real root, or no real roots. (a) 2x2 + 4x − 6 = 0 (b) 3x2 – 6x + 8 =0 (c) x2 + 2x + 1 =0
Solution:
(a) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 2, b = 4, c = -6 Substituting the values in the discriminant b2 – 4ac, we get = (4)2 – 4 × 2 × -6 = 64 Since the value of discriminant is positive (> 0), hence the equation 2x2 + 4x − 6 = 0 has two real roots (b) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 3, b = -6, c = 8 Substituting the values in the discriminant b2 – 4ac, we get = (-6)2 – 4 × 3 × 8 = -60 Since the value of the discriminant is negative (< 0), the equation 3x2 – 6x + 8 =0 has no real solutions but two different complex roots. (c) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 1, b = 2, c = 1 Substituting the values in the discriminant b2 – 4ac, we get = (2)2 – 4 × 1 × 1 = 0 Since the value of the discriminant is zero (= 0), the equation x2 + 2x + 1 =0 has one real root.
