If the equations ${x^2} + ax + bc = 0\,and\,{x^2} + bx + ca = 0$ have a common root and if a, b and c are non-zero distinct real numbers, then their roots satisfy the equation:
A. ${x^2} + x + abc = 0\,\,$
B. ${x^2} - \left( {a + b} \right)x + ab = 0\,\,$
C. ${x^2} + \left( {a + b} \right)x + ab = 0\,\,$
D. ${x^2} + x + ab = 0\,\,$
E. ${x^2} + abx + abc = 0\,\,$

Question

If the equations ${x^2} + ax + bc = 0\,and\,{x^2} + bx + ca = 0$ have a common root and if a, b and c are non-zero distinct real numbers, then their roots satisfy the equation:
A. ${x^2} + x + abc = 0\,\,$
B. ${x^2} - \left( {a + b} \right)x + ab = 0\,\,$
C. ${x^2} + \left( {a + b} \right)x + ab = 0\,\,$
D. ${x^2} + x + ab = 0\,\,$
E. ${x^2} + abx + abc = 0\,\,$

Vedantu Content Team · Accepted Answer

Hint: We have been given with two equations, to find a common root we will solve both the equations.

We will subtract the second equation from the first one,
$\left( {a - b} \right)x + c\left( {b - a} \right) = 0$
$ \Rightarrow \left( {a - b} \right)\left( {x - c} \right) = 0$
is the common root.
Thus, the roots of ${x^2} + ax + bc = 0$ are b and c and that of ${x^2} + bx + ca = 0$ are c and a.
If roots are b and a, then the equation will be,
${x^2} - \left( {a + b} \right)x + ab = 0\,$
Answer is option B

Note: In this question, we solved the equations first and then found the common root and then form the equation using the roots.

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