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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

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.

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