Table of Contents

Last modified on August 3rd, 2023

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 ax^{2} + bx + c = 0, b^{2} – 4ac is the discriminant (D), as shown in the diagram below.

Thus, to find the discriminant of a quadratic equation, follow the following steps:

**Step 1**: Compare the given quadratic equation with its standard form ax^{2} + bx + c = 0 and find the values of a, b and c

**Step 2**: Substitute the values in the discriminant b^{2} – 4ac to get the result

Let us find the discriminant of the quadratic equation x^{2} + 10x + 16 = 0

Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0, we get a = 1, b = 10, c = 16

Substituting the values in the discriminant b^{2} – 4ac, we get

= (10)^{2} – 4 Ã— 1 Ã— 16

= 100 â€“ 64

= 36

Thus, the discriminant of the quadratic equation x^{2} + 10x + 16 = 0 is 36

**What is the discriminant of the quadratic equation 3 – 4x – 6x ^{2} = 0?**

Solution:

Rearranging the equation to -6x^{2} – 4x + 3 = 0

Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0 we get, a = -6, b = -4, c = 3

Substituting the values in the discriminant b^{2} – 4ac, we get

= (-4)^{2} – 4 Ã— -6 Ã— 3

Â = -56

Thus, the discriminant of the quadratic equation 3 – 4x – 6x^{2} = 0 is 36

**Determine the discriminant for the quadratic equation -3x ^{2 }+ 4x + 1 = 0**

Solution:

Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0 we get, a = -3, b = 4, c = 1

Substituting the values in the discriminant b^{2} – 4ac, we get

= (4)^{2} – 4 Ã— -3 Ã— 1

Â = 28

Thus, the discriminant of the quadratic equation -3x^{2 }+ 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 b
^{2}– 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 b
^{2}– 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 b
^{2}– 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) 2x ^{2}Â + 4x âˆ’ 6 = 0**

Solution:

(a) Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0, we get a = 2, b = 4, c = -6

Substituting the values in the discriminant b^{2} – 4ac, we get

= (4)^{2} – 4 Ã— 2 Ã— -6

= 64

Since the value of discriminant is positive (> 0), hence the equation 2x^{2}Â + 4x âˆ’ 6 = 0 has two real roots

(b) Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0, we get a = 3, b = -6, c = 8

Substituting the values in the discriminant b^{2} – 4ac, we get

= (-6)^{2} – 4 Ã— 3 Ã— 8

= -60

Since the value of the discriminant is negative (< 0), the equation 3x^{2} – 6x + 8 =0 has no real solutions but two different complex roots.

(c) Comparing the given quadratic equation with its standard form ax^{2} + bx + c = 0, we get a = 1, b = 2, c = 1

Substituting the values in the discriminant b^{2} – 4ac, we get

= (2)^{2} – 4 Ã— 1 Ã— 1

= 0

Since the value of the discriminant is zero (= 0), the equation x^{2} + 2x + 1 =0 has one real root.

Last modified on August 3rd, 2023