Difference between the corresponding roots of x^2+ax+b =0 and x^2 + bx +a =0 is same and a≠b then :A) a + b + 4 = 0B) a + b - 4 = 0C) a - b - 4 = 0D) a- b + 4 = 0

To tackle this problem, we need to analyze the roots of the given quadratic equations and their relationship. We have two equations: \(x^2 + ax + b = 0\) and \(x^2 + bx + a = 0\). The condition states that the difference between the corresponding roots of these two equations is the same, and we also know that \(a \neq b\).

Calculating the Roots

The roots of a quadratic equation of the form \(x^2 + px + q = 0\) can be found using the quadratic formula:

Roots = \(\frac{-p \pm \sqrt{p^2 - 4q}}{2}\)

Roots of the First Equation

For the equation \(x^2 + ax + b = 0\), we can identify:

• p = a
• q = b

The roots are:

Root 1 = \(\frac{-a + \sqrt{a^2 - 4b}}{2}\)

Root 2 = \(\frac{-a - \sqrt{a^2 - 4b}}{2}\)

Roots of the Second Equation

For the second equation \(x^2 + bx + a = 0\), we have:

• p = b
• q = a

The roots are:

Root 1' = \(\frac{-b + \sqrt{b^2 - 4a}}{2}\)

Root 2' = \(\frac{-b - \sqrt{b^2 - 4a}}{2}\)

Finding the Differences Between Roots

The difference between the roots of the first equation is:

Difference 1 = \(\sqrt{a^2 - 4b}\)

And for the second equation, the difference is:

Difference 2 = \(\sqrt{b^2 - 4a}\)

Setting Up the Equation

According to the problem, these differences are equal:

\(\sqrt{a^2 - 4b} = \sqrt{b^2 - 4a}\)

Squaring both sides to eliminate the square roots gives us:

a^2 - 4b = b^2 - 4a

Rearranging the Equation

Now, let's rearrange this equation:

• Move all terms to one side: a^2 - b^2 + 4a - 4b = 0
• This can be factored as: (a - b)(a + b) + 4(a - b) = 0

Factoring out \(a - b\), we get:

(a - b)(a + b + 4) = 0

Considering the Conditions

Since we know \(a \neq b\), we cannot have \(a - b = 0\). Therefore, we must have:

a + b + 4 = 0

Final Answer

Thus, the condition that holds true is:

A) a + b + 4 = 0

This analysis shows the importance of manipulating equations and understanding quadratic roots. If you have further questions about this topic or want to delve deeper into any specific part, feel free to ask!