State whether the following quadratic equations have two distinct real roots.
Justify your answer.
$3 x^{2} - 4 x + 1 = 0$

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Solution

Nature of roots.
The discriminants of a quadratic equation $a x^{2} + b x + c = 0$ is given by,
$D = b^{2} - 4 a c$
If $D > 0$ , the roots are distinct and real.
If $D = 0$ , the roots are equal and real.
If $D < 0$ , the roots will be imaginary.
The discriminant of the given quadratic equation $3 x^{2} - 4 x + 1 = 0$ is computed as:
$∴ D = {(- 4)}^{2} - 4 \times 3 \times 1 \Rightarrow D = 16 - 12 \Rightarrow D = 4 \Rightarrow D > 0$
Since the discriminant is greater than zero.
Hence, the roots are distinct and real.

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