Question
Question: If \[{x^2} + ax + bc = 0\] and \[{x^2} + bx + ca = 0\] (\[a \ne b\] and \[c \ne 0\]) have a common r...
If x2+ax+bc=0 and x2+bx+ca=0 (a=b and c=0) have a common root, then a+b+c=
A.0
B.1
C.ab+bc+ca
D.3abc
Solution
Here, we have to find the sum of the roots. First, we will use the condition if two roots have a common root, to find the common root. Then by using the condition we will find the sum of the roots. A quadratic equation is an equation of a variable with the highest degree is 2.
Formula Used:
The difference of the square of two numbers is given by a2−b2=(a+b)(a−b)
Complete step-by-step answer:
Let us consider two quadratic equations a1x2+b1x+c1=0 and a2x2+b2x+c2=0.
If the two equations have a common root, then we have the condition that,
b1c1−c2b2x2=a1c2−c1a2−x=a1b2−b1a21 ………………………..(1)
Now, consider the equation x2+ax+bc=0 and x2+bx+ca=0.
We will now use the condition in equation (1) for the given equations.
Now, Considering the last two terms
a1c2−c1a2−x=a1b2−b1a21
By substituting the values of the coefficients and the constant term, we get
⇒1⋅ca−bc⋅1−x=1⋅b−a⋅11
By taking out the common terms, we get
⇒ca−bc−x=b−a1
By cancelling the terms, we get
⇒c(a−b)−x=(−1)(a−b)1
⇒cx=1
By cross- multiplying the equation, we get
⇒x=c ………………………………………………………………………………..(2)
Now, considering the first two terms, we get
b1c1−c2b2x2=a1c2−c1a2−x
By cancelling both the terms, we get
⇒b1c1−c2b2x=a1c2−c1a2−1
By substituting the values of the coefficients and the constant term, we get
⇒a⋅ac−bc⋅bx=1⋅ca−bc⋅1−1
By multiplying the terms, we get
⇒a2c−b2cx=ca−bc−1
By taking out the common terms, we get
⇒c(a2−b2)x=c(a−b)−1
By substituting equation (2) in the above equation, we get
⇒(a2−b2)c=(a−b)−1
Using the algebraic identity a2−b2=(a+b)(a−b), we get
⇒(a+b)(a−b)c=(a−b)−1
By cancelling out the common terms, we get
⇒(a+b)c=−1
By cross-multiplying, we get
⇒c=−1(a+b)
⇒c=−a−b
By rewriting the equation, we get
⇒a+b+c=0
Therefore, if x2+ax+bc=0 and x2+bx+ca=0 have a common root, then a+b+c=0.
Thus, option (A) is the correct answer.
Note: A quadratic equation is an equation, which has the highest degree of variable as 2 and has two solutions. We should remember the use of the condition if the quadratic equations have a common root to find the value of the common root. We also have a condition that if the two quadratic equations have a common root, then the sum of the roots is always zero whatever be the quadratic equation.