Without solving, examine the nature of the roots of the equation $2 x^{2} - 6 x + 3 = 0$ .

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Solution

Finding the nature of roots of the equation $2 x^{2} - 6 x + 3 = 0$ .
Compare the given quadratic equation $2 x^{2} - 6 x + 3 = 0$ with the standard quadratic equation $a x^{2} + b x + c = 0$ .
$\begin{array}{rcl} a & = & 2 \\ b & = & - 6 \\ c & = & 3 \end{array}$
The discriminant of the quadratic equation is,
$D = b^{2} - 4 a c$
Determine the value of $D$ .
$\begin{array}{rcl} D & = & {(- 6)}^{2} - 4 \times 2 \times 3 \\ = & 36 - 24 \\ = & 12 > 0 \end{array}$
Since, $D > 0$ , the equation has two real and distinct roots.
Thus, the quadratic equation $2 x^{2} - 6 x + 3 = 0$ has two real and distinct roots.

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