Question

Find the discriminant of the quadratic equation $2x^2-6x+3=0$, 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 $△=b^2-4ac$
Check which one of the following conditions are satisfied by the discriminant and answer the question accordingly:
1)$△>0\Rightarrow$Two unequal and real roots
2) $△=0\Rightarrow$Two equal and real roots
3) $△<0\Rightarrow$Two complex roots

Complete step by step solution: We are given a quadratic equation $2x^2-6x+3=0$
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
$x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
The expression $b^2-4ac$ is called the discriminant of the quadratic equation and is denoted by $△$.
Also, as the expression is a real number, the discriminant will be either zero, or negative or positive.
Comparing the given quadratic equation $2x^2-6x+3=0$ with the general form, we get
a = 2, b = -6, and c = 3.
Therefore,
$$\begin{gathered} △=b^2-4ac \\ =(-6{)}^2-4\times 2\times 3 \\ =36-24 \\ =12 \end{gathered}$$
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.