Question

The sum of a number and its square root is 110. What is the number?

(a) 100

(b) 102

(c) 99

(d) 98

This question was addressed to me in unit test.

This is a very interesting question from Solution of Quadratic Equation by Squaring Method topic in chapter Quadratic Equation of Mathematics – Class 10

Answer 1

Correct option is (a) 100

Best explanation: Let the number be x. It square root will be √x

Sum of the number and its square root is 110.

x+√x=110

Let √x=y, x=y^2

The equation becomes,

y^2+y=110

Adding \(\frac {b^2}{4}\) on both sides, where b=1

y^2 + y + \(\frac {1^2}{4}=\frac {1^2}{4}\) + 110

y^2 + y + \(\frac {1}{4}=\frac {1}{4}\) + 110

y^2 + y + \(\frac {1}{4}=\frac {441}{4}\)

\((y+\frac {1}{2})\)^2 = \((\frac {21}{2})\)^2

y + \(\frac {1}{2}\) = ±\(\frac {21}{2}\)

y = \(\frac {21}{2}-\frac {1}{2}=\frac {20}{2}\) = 10 and y = \(\frac {-20}{2}-\frac {1}{2}=\frac {-21}{2}\) = -10.5

x = y^2 = 10^2 = 100 and x = y^2 = -10.5^2 = 110.25

The numbers are 100 and 110.25.