Question

If \[\{ 8,2\} \] are the roots of \[{x^2} + ax + \beta = 0\] and \[\{ 3,3\} \] are the roots of \[{x^2} + \alpha x + b = 0\], then the roots of the equation \[{x^2} + ax + b = 0\]are:
(A). \[1, - 1\]
(B). \[ - 9,2\]
(C). \[ - 8, - 2\]
(D). \[9,1\]

Answer 1

- Hint: Before finding the roots of the given equation, at first, we will find the value of \[a,b\].
Let us consider \[p,q\] be the roots of a quadratic equation. Then, the equation can be written as,
\[{x^2} - (p + q)x + pq = 0\]

Complete step-by-step solution -
It is given that, \[\{ 8,2\} \] are the roots of \[{x^2} + ax + \beta = 0\]. Also, given that, \[\{ 3,3\} \] are the roots of \[{x^2} + \alpha x + b = 0\].
We have to find the roots of the equation \[{x^2} + ax + b = 0\].
Before finding the roots of the given equation, at first, we will find the value of \[a,b\].
Let us consider, \[p,q\] be the roots of a quadratic equation. Then, the equation can be written as,
\[{x^2} - (p + q)x + pq = 0\]
Since, \[\{ 8,2\} \] are the roots of a quadratic equation, the equation could be, \[{x^2} - (8 + 2)x + 8 \times 2 = 0\]
Simplifying we get, the equation is, \[{x^2} - 10x + 16 = 0\]
Comparing the equations, \[{x^2} - 10x + 16 = 0\] and \[{x^2} + ax + \beta = 0\] we get,
\[a = - 10,\beta = 16\]
Similarly,
Since, \[\{ 3,3\} \] are the roots of a quadratic equation, the equation could be, \[{x^2} - (3 + 3)x + 3 \times 3 = 0\]
Simplifying we get, the equation is, \[{x^2} - 6x + 9 = 0\]
Comparing the equations, \[{x^2} - 6x + 9 = 0\] and \[{x^2} + \alpha x + b = 0\] we get,
\[\alpha = - 6,b = 9\]
Substitute the value of \[a = - 10,b = 9\] in the equation \[{x^2} + ax + b = 0\]we get,
The equation as: \[{x^2} - 10x + 9 = 0\]
Now, we will apply the middle term factor method to find out the roots.
So, the equation can be written as,
\[{x^2} - (9 + 1)x + 9 = 0\]
Simplifying we get,
\[{x^2} - 9x - x + 9 = 0\]
Simplifying again we get,
\[(x - 9)(x - 1) = 0\]
Hence, the roots are \[9,1\]

Hence, the correct option is (D) \[9,1\]

Note:
To find the roots of the quadratic equation we can apply Sreedhar Acharya’s formula instead of middle term factorization.
It states that, if \[a{x^2} + bx + c = 0\] be a quadratic equation then its roots are,
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
We will find the same answer as above.