Question:
If the difference between the roots of
\[
x^2+ax+b=0
\]
and that of the roots of
\[
x^2+bx+a=0
\]
is same and \(a\neq b\), then
If the difference between the roots of
\[
x^2+ax+b=0
\]
and that of the roots of
\[
x^2+bx+a=0
\]
is same and \(a\neq b\), then
Show Hint
For quadratic equations, the difference between roots depends on the discriminant:
\[
\text{Difference of roots}=\sqrt{D}
\]
when the coefficient of \(x^2\) is \(1\).
Updated On: Jun 22, 2026
- \(a-b-4=0\)
- \(a-b+4=0\)
- \(a+b+4=0\)
- \(a+b-4=0\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Recall the formula for the difference of roots.
For a quadratic equation \[ Ax^2+Bx+C=0, \] the difference between the roots is \[ \frac{\sqrt{B^2-4AC}}{A} \] Since the coefficient of \(x^2\) is \(1\) in both equations, the difference of roots depends only on the discriminant.
Step 2: Find the discriminant of the first equation.
For \[ x^2+ax+b=0, \] the discriminant is \[ D_1=a^2-4b \] Hence, the difference between the roots is \[ \sqrt{a^2-4b} \]
Step 3: Find the discriminant of the second equation.
For \[ x^2+bx+a=0, \] the discriminant is \[ D_2=b^2-4a \] Hence, the difference between the roots is \[ \sqrt{b^2-4a} \]
Step 4: Use the given condition.
Given that the differences between the roots are same, \[ \sqrt{a^2-4b} = \sqrt{b^2-4a} \] Squaring both sides, \[ a^2-4b=b^2-4a \] Rearranging, \[ a^2-b^2+4a-4b=0 \] Factorizing, \[ (a-b)(a+b)+4(a-b)=0 \] Taking \((a-b)\) common, \[ (a-b)(a+b+4)=0 \] Given, \[ a\neq b, \] therefore, \[ a+b+4=0 \]
Step 5: Final conclusion.
Hence, \[ \boxed{a+b+4=0} \] which corresponds to option (3).
For a quadratic equation \[ Ax^2+Bx+C=0, \] the difference between the roots is \[ \frac{\sqrt{B^2-4AC}}{A} \] Since the coefficient of \(x^2\) is \(1\) in both equations, the difference of roots depends only on the discriminant.
Step 2: Find the discriminant of the first equation.
For \[ x^2+ax+b=0, \] the discriminant is \[ D_1=a^2-4b \] Hence, the difference between the roots is \[ \sqrt{a^2-4b} \]
Step 3: Find the discriminant of the second equation.
For \[ x^2+bx+a=0, \] the discriminant is \[ D_2=b^2-4a \] Hence, the difference between the roots is \[ \sqrt{b^2-4a} \]
Step 4: Use the given condition.
Given that the differences between the roots are same, \[ \sqrt{a^2-4b} = \sqrt{b^2-4a} \] Squaring both sides, \[ a^2-4b=b^2-4a \] Rearranging, \[ a^2-b^2+4a-4b=0 \] Factorizing, \[ (a-b)(a+b)+4(a-b)=0 \] Taking \((a-b)\) common, \[ (a-b)(a+b+4)=0 \] Given, \[ a\neq b, \] therefore, \[ a+b+4=0 \]
Step 5: Final conclusion.
Hence, \[ \boxed{a+b+4=0} \] which corresponds to option (3).
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