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Which of the following quadratic expression can be expressed as product of real linear factors?

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x^{2} - 2 x + 3

3 x^{2} - \sqrt{2} x - \sqrt{3}

\sqrt{2} x^{2} - \sqrt{5} x + 3

None of these

answer is B.

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Detailed Solution

Real linear factors can be used to describe a quadratic expression if their discriminant is larger than or equal to zero.
For the real linear factors of standard quadratic equation,

a x^{2} + bx + c = 0

D = (b)^{2} - 4 ac \geq 0

Option (1),

x^{2} - 2 x + 3

Discriminant of this equation,

D = (- 2)^{2} - 4 \times 1 \times 3

= 4 - 12 < 0

Therefore, the quadratic expression it cannot be expressed of real linear factors.
Option (2),

3 x^{2} - \sqrt{2} x - \sqrt{3}

Discriminant of this equation,

D = (- \sqrt{3})^{2} - \sqrt{2} x - \sqrt{3}

= 2 + 12 \sqrt{3} > 0

Thus, the quadratic expression it can be expressed of real linear factors.
Option (3)

\sqrt{2} x^{2} - \sqrt{5} x + 3

Discriminant of this equation,

D = (- \sqrt{5})^{2} - 4 \times (\sqrt{2}) \times 3

D = 5 - 12 \sqrt{2} < 0

So, the quadratic expression it cannot be expressed of linear factors.
Hence, the correct option is 2.

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Which of the following quadratic expression can be expressed as product of real linear factors?

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