Solving for Common Root in $(1)$ and $(2)$

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In summary, a common root in mathematics is a value that satisfies two or more equations simultaneously. To solve for a common root, you can use the method of substitution or elimination. Finding a common root is important for determining the point of intersection between two equations and can also indicate if the equations are equivalent. It is possible for two equations to have more than one common root, but if they do not have a common root, it means that there is no solution that satisfies both equations.
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$(a-1)x^2-(a^2+2)x+(a^2+2a)=0----(1)\\
(b-1)x^2-(b^2+2)x+(b^2+2b)=0----(2)
$
if $(1)$ and $(2)$ have one root in common ,
(here $a,b\in N$ ,$a\neq b,\,\, and \,\, a>1,b>1$)
find value of :
$\dfrac{a^a+b^b}{a^{-b}+b^{-a}}$
 
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Albert said:
$(a-1)x^2-(a^2+2)x+(a^2+2a)=0----(1)\\
(b-1)x^2-(b^2+2)x+(b^2+2b)=0----(2)
$
if $(1)$ and $(2)$ have one root in common ,
(here $a,b\in N$ ,$a\neq b,\,\, and \,\, a>1,b>1$)
find value of :
$\dfrac{a^a+b^b}{a^{-b}+b^{-a}}$
hint
(1) and (2) can be factorized
 

FAQ: Solving for Common Root in $(1)$ and $(2)$

What is a common root in mathematics?

A common root in mathematics refers to a value that satisfies two or more equations simultaneously. In other words, it is a value that, when substituted into each equation, makes both equations true.

How do you solve for a common root in two equations?

To solve for a common root in two equations, you can use the method of substitution or elimination. In the method of substitution, you solve one equation for one variable and then substitute that value into the other equation. In the method of elimination, you manipulate the equations to eliminate one variable and then solve for the remaining variable.

What is the importance of finding a common root?

Finding a common root is important because it allows us to find the point of intersection between two equations, which can be useful in solving real-world problems. It also helps us to determine if two equations are equivalent or if they have a unique solution.

Can two equations have more than one common root?

Yes, two equations can have more than one common root. This means that there can be multiple values that satisfy both equations simultaneously.

What if the two equations do not have a common root?

If the two equations do not have a common root, it means that there is no value that satisfies both equations simultaneously. This could indicate that the equations are parallel or have no intersection point.

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