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Standard X

Mathematics

Solving a Quadratic Equation by Completion of Squares Method

The number of...

Question

The number of values of $k$ for which $(x^{2} - (k - 2) x + k^{2}) (x^{2} + k x + (2 k - 1))$ is a perfect square

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Solution

The correct option is B $1$
Given expression
$(x^{2} - (k - 2) x + k^{2}) (x^{2} + k x + (2 k - 1))$
For the expression to be a perfect square, the two quadratic factors should have coefficients of $x$ and constant terms equal.
$\Rightarrow - (k - 2) = k$ and $k^{2} = 2 k - 1$
$\Rightarrow k = 1$ and $(k - 1)^{2} = 0$
$\Rightarrow k = 1$
Hence, the number of values of $k$ is 1.

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The number of values of k for which (x2−(k−2)x+k2)(x2+kx+(2k−1)) is a perfect square

The correct option is B 1Given expression (x2−(k−2)x+k2)(x2+kx+(2k−1))For the expression to be a perfect square, the two quadratic factors should have coefficients of x and constant terms equal. ⇒−(k−2)=k and k2=2k−1⇒k=1 and (k−1)2=0⇒k=1Hence, the number of values of k is 1.

The number of values of $k$ for which $(x^{2} - (k - 2) x + k^{2}) (x^{2} + k x + (2 k - 1))$ is a perfect square