Question

If 2 is a root of the quadratic equation \[3{x^2} + px - 8 = 0\] and the quadratic equation \[4{x^2} - 2px + k = 0\] has equal roots. The value of k is---.

Answer 1

Hint: If \[\alpha \]Is a root of any quadratic equation \[a{x^2} + bx + c = 0\] then, it will satisfy the equation i.e.
\[a{\left( \alpha \right)^2} + b\left( \alpha \right) + c = 0\]
So, 2 will satisfy the quadratic \[3{x^2} + px - 8 = 0\]. Hence, find 'p' using this condition. Now, use the property of any quadratic equation \[a{x^2} + bx + c = 0\], which has equal roots then discriminant of this quadratic is 0 i.e.
\[D = {b^2} - 4ac = 0\]
So, put \[D = 0\] of the second quadratic to get the value of k.

Complete step-by-step answer:
As we know, if \[\alpha \] is a root of any equation \[a{x^2} + bx + c = 0\] , then \['\alpha '\] will satisfy this equation i.e.
\[a{\left( \alpha \right)^2} + b\left( \alpha \right) + c = 0\]
Now, as it is given that 2 is a root of the quadratic equation \[3{x^2} + px - 8 = 0\] , then '2' will satisfy the given equation. So, we get
\[\begin{array}{l}3{\left( 2 \right)^2} + p\left( 2 \right) - 8 = 0\\12 + 2p - 8 = 0\end{array}\]
Or \[2p + 4 = 0\]
Or \[2p = - 4\]


On dividing the above equation by 2, we get:
\[\dfrac{{2p}}{2} = \dfrac{{ - 4}}{2}\]
\[p = - 2\] . . . . . . . . . . . (i)
Now, we are also given that the quadratic equation \[4{x^2} - 2px + k = 0\] has equal roots. So, as we know any quadratic equation \[a{x^2} + bx + c = 0\] will have equal roots if:
\[D = {b^2} - 4ac = 0\]
Where, D is the discriminant of the quadratic \[a{x^2} + bx + c = 0\]
So, the discriminant of \[4{x^2} - 2px + k = 0\] will be 0 as well. Hence, we can use equation (ii) and get the relation as:
\[\begin{array}{l}{\left( { - 2p} \right)^2} - 4 \times 4 \times k = 0\\4{p^2} - 16k = 0\end{array}\]
Or \[4{p^2} = 16k\]

Now, we can put value of 'p' as '-2' using equation (i) to the above expression and hence, we get:
\[\begin{array}{l}4{\left( { - 2} \right)^2} = 16k\\16 = 16k\end{array}\]
Or \[k = 1\]
Hence, we get the value of k as 1.
So, \[k = 1\]is the answer to the problem.


Note: One does not need to find the exact roots of the given quadratic equations. As one may go wrong, if he/she tries to find the roots of quadratic \[3{x^2} + px - 8 = 0\] and try to put one of the roots as '2', which will be the wrong approach and complex for some of the students.
Any quadratic \[a{x^2} + bx + c = 0\] has real and equal roots for:
\[D = {b^2} - 4ac = 0\]

And discriminant D of any quadratic will decide the property of roots by following criteria
\[D = {b^2} - 4ac \ge 0\] , Real roots
\[D = {b^2} - 4ac < 0\] , Unreal roots
Hence, use the above concept for future reference.