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

x^{2} - 2 x + 3

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

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

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None of these

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Solution

The correct option is D $3 x^{2} - \sqrt{2} x - \sqrt{3}$
For real roots $b^{2} - 4 a c \geq 0$
Hence
A)
$x^{2} - 2 x + 3 = 0$
$b^{2} - 4 a c$
$= 4 - 4 (3)$
$= - 8$
Hence non-real roots.
B)
$3 x^{2} - \sqrt{2} x - \sqrt{3} = 0$
$b^{2} - 4 a c$
$= 2 - 4 (- \sqrt{3} .3)$
$= 2 + 12 \sqrt{3}$
Hence real roots.
C)
$\sqrt{2} x^{2} - \sqrt{5} x + 3 = 0$
Hence
$b^{2} - 4 a c$
$= 5 - 4 (3. \sqrt{2})$
$= 5 - 12 \sqrt{2}$
Hence non-real roots.
Thus the answer is B.

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Question 2
Which of the following is not a quadratic equation?
(a) $2 (x - 1)^{2} = 4 x^{2} - 2 x + 1$
(b) $2 x - x^{2} = x^{2} + 5$
(c) ${(\sqrt{2} x + \sqrt{3})}^{2} = 3 x^{2} - 5 x$
(d) $(x^{2} + 2 x)^{2} = x^{4} + 3 + 4 x^{3}$

Question 2
Which of the following is not a quadratic equation?
(a) $2 (x - 1)^{2} = 4 x^{2} - 2 x + 1$
(b) $2 x - x^{2} = x^{2} + 5$
(c) ${(\sqrt{2} x + \sqrt{3})}^{2} = 3 x^{2} - 5 x$
(d) $(x^{2} + 2 x)^{2} = x^{4} + 3 + 4 x^{3}$

Q. Write the discriminant of the following quadratic equations:

(i) 2x² − 5x + 3 = 0
(ii) x² + 2x + 4 = 0
(iii) (x − 1) (2x − 1) = 0
(iv) x² − 2x + k = 0, k ∈ R
(v)

\sqrt{3} x^{2} + 2 \sqrt{2} x - 2 \sqrt{3} = 0

(vi) x² − x + 1 = 0
(vii) (x + 5)² = 2(5x – 3)

Q. Find the range of following quadratic expression.

f (x) = - x^{2} + 2 x + 3

\forall x \in R

Q. The expression x⁴ + 4 can be factorized as

(a) (x² + 2x + 2) (x² − 2x + 2)

(b) (x² + 2x + 2) (x² + 2x + 2)

(c) (x² − 2x − 2) (x²− 2x + 2)

(d) (x² + 2) (x² − 2)

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