Question

If one root of 8x² -2x + k = 0 is reciprocal of other. The value of k is:​

Answer 1

$\bigstar$ Explanation $\bigstar$

$\leadsto$ Solution:-

We know that if $\rm ax^2 + bx + c= 0$ then,

$\rm x = \dfrac{-b\pm\sqrt{b^2 - 4ac}}{2a}$

In the given question,

$\rm 8x^2 - 2x + k$ = 0

Therefore by using the above formula,

$\rm x = \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(8)(k)} }{2(8)} = \dfrac{2\pm\sqrt{4 - 32k}}{16}$

According to the question,

One root is reciprocal of the other root.

Therefore,

$\dfrac{2+\sqrt{4 - 32k}}{16} = \dfrac{1}{\dfrac{2-\sqrt{4 - 32k}}{16}} \\\\\dfrac{2+\sqrt{4 - 32k}}{16} = \dfrac{16}{2+\sqrt{4 - 32k}}\\\\16\times16 = (2+\sqrt{4 - 32k})(2-\sqrt{4 - 32k})$

We know that $\rm (a+b)(a-b) = a^2-b^2$

Therefore,

$\rm 256 = 2^2 - [\sqrt{4-32k}]^2$

$\rm 256 = 4 - 4 + 32k\implies 256 = 32k \implies k = 8$

$\leadsto$ Important Concepts related to quadratic equations:-

i) If $\rm ax^2 + bx+ c = 0$, then

$\rm x = \dfrac{-b\pm\sqrt{b^2 - 4ac}}{2a}$

ii) The discriminant of a quadratic equation is $\rm b^2 - 4ac$,

If  discriminant > 0, then the roots are real and distinct and unequal

If discriminant = 0, then roots are real and equal

If discriminant < 0, then the roots are imaginary and distinct and unequal.

iii) If the roots of a quadratic equation are $\alpha$ and $\beta$,

The sum of the roots = $\alpha +\beta$ = $\rm \frac{-b}{a}$

The product of the roots = $\alpha \beta$ = $\rm \frac{c}{a}$