Question

if x^2+ax+bc=0 and x^2+bx+ca=0 have a common root ,then a+b+c is
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Answer 1

Answer:

Hint: 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)

Explanation:

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,

x2b1c1−c2b2=−xa1c2−c1a2=1a1b2−b1a2

………………………..(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

−xa1c2−c1a2=1a1b2−b1a2

By substituting the values of the coefficients and the constant term, we get

⇒−x1⋅ca−bc⋅1=11⋅b−a⋅1

By taking out the common terms, we get

⇒−xca−bc=1b−a

By cancelling the terms, we get

⇒−xc(a−b)=1(−1)(a−b)

⇒xc=1

By cross- multiplying the equation, we get

⇒x=c

………………………………………………………………………………..(2)

Now, considering the first two terms, we get

x2b1c1−c2b2=−xa1c2−c1a2

By cancelling both the terms, we get

⇒xb1c1−c2b2=−1a1c2−c1a2

By substituting the values of the coefficients and the constant term, we get

⇒xa⋅ac−bc⋅b=−11⋅ca−bc⋅1

By multiplying the terms, we get

⇒xa2c−b2c=−1ca−bc

By taking out the common terms, we get

⇒xc(a2−b2)=−1c(a−b)

By substituting equation (2)

in the above equation, we get

⇒c(a2−b2)=−1(a−b)

Using the algebraic identity a2−b2=(a+b)(a−b)

, we get

⇒c(a+b)(a−b)=−1(a−b)

By cancelling out the common terms, we get

⇒c(a+b)=−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