If arithmetic mean and geometric mean of roots of a quadratic equation are 8 and 5 respectively, then the quadratic equation is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x^2-16x+25=0\)
A quadratic equation with roots \(a\) and \(b\) is \(x^2-(a+b)x+ab=0\), so only the sum and the product of the roots are needed.
Arithmetic mean \(\displaystyle =\frac{a+b}{2}=8\), so the sum \(a+b=16\)
Geometric mean \(=\sqrt{ab}=5\), so the product \(ab=25\)
The equation is \(x^2-16x+25=0\)
Tip: The arithmetic mean of the two roots is half their sum, so the coefficient of \(x\) is twice the mean (16), not the mean itself; Option (2) uses 8.