Question:medium

The mean proportional of the two positive numbers obtained by subtracting 77 from a positive number and adding 11 to the same number is 33. What is the number?

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Write the mean proportional condition as \(\sqrt{(x-77)(x+11)}=33\), square both sides, and solve the resulting quadratic, keeping only the positive root that keeps both parts positive.
Updated On: Jul 20, 2026
  • 154
  • 143
  • 121
  • 99
  • 88
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The Correct Option is

Solution and Explanation

An alternative approach uses factoring instead of the quadratic formula.
From \((x-77)(x+11)=1089\), expand to get \(x^2-66x-1936=0\).
We need two numbers that multiply to \(-1936\) and add to \(-66\). Trying \(-88\) and \(22\): product \(=-88\times22=-1936\) and sum \(=-88+22=-66\). Both match, so the equation factors as:
$$x^2-66x-1936=(x-88)(x+22)=0$$
This gives \(x=88\) or \(x=-22\). Since the number is positive, \(x=88\).
Quick check: \(x-77=11\), \(x+11=99\), and \(11\times99=1089=33^2\), confirming the mean proportional is indeed 33.\[\boxed{88}\]
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