Question

Find the discriminant of the quadratic equation $2{x^2} - 6x + 3 = 0$, and hence find the nature of its roots.

Answer 1

Hint: Compare the equation with the general form $a{x^2} + bx + c = 0$ and find the values of a, b, and c to compute the discriminant $\vartriangle = {b^2} - 4ac$
Check which one of the following conditions are satisfied by the discriminant and answer the question accordingly:
1)$\vartriangle > 0 \Rightarrow $Two unequal and real roots
2) $\vartriangle = 0 \Rightarrow $Two equal and real roots
3) $\vartriangle < 0 \Rightarrow $Two complex roots

Complete step by step solution: We are given a quadratic equation $2{x^2} - 6x + 3 = 0 $
There are three parts to this question
First - Find the discriminant of the given equation
Second - Determine the nature of the roots using the discriminant

Consider the general form of a quadratic equation $a{x^2} + bx + c = 0$ where a, b, c are real numbers and ‘a’ is a non-zero constant.
We know that the roots of this equation are computed by using the formula
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
The expression ${b^2} - 4ac$ is called the discriminant of the quadratic equation and is denoted by $\vartriangle $.
Also, as the expression is a real number, the discriminant will be either zero, or negative or positive.
Comparing the given quadratic equation $2{x^2} - 6x + 3 = 0$ with the general form, we get
a = 2, b = -6, and c = 3.
Therefore,
$ \vartriangle = {b^2} - 4ac \\
   = {( - 6)^2} - 4 \times 2 \times 3 \\
   = 36 - 24 \\
   = 12 \\ $
Thus, the discriminant of the given quadratic equation is 12.
Based on the discriminant of a quadratic equation, we can have the following answers:
1) if the discriminant is positive, then we will get a pair of real solutions; i.e. there will be two distinct (unequal) roots and they will be real numbers
2) if the discriminant is zero, then we get only one solution which will be a real number; i.e. there will be two equal roots and the number will be a real number
3) if the discriminant is negative, then we get a pair of complex solutions; i.e. there will be two roots which will be complex numbers

Now, the discriminant of the given quadratic equation is 12 i.e. a positive number and hence satisfies the first condition.
Therefore, the roots of the given equation will be distinct and real.

Note: As the question says ‘hence’ determine the nature of the roots, there is no need to compute the roots; we must only use the discriminant to draw our conclusions.