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Question
Product of Pragati’s age 2 years ago and 3 years hence is 84. Find her present age.
Solution 1
Let Pragati's present age be x years.
Her age 2 years ago = x - 2
Her age 3 years hence = x + 3
Product of Pragati’s age 2 years ago and 3 years hence is 84.
\[\left( x - 2 \right)\left( x + 3 \right) = 84\]
\[ \Rightarrow x^2 + x - 6 = 84\]
\[ \Rightarrow x^2 + x - 90 = 0\]
\[x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[ \Rightarrow x = \frac{- 1 \pm \sqrt{1^2 - 4 \times 1 \times \left( - 90 \right)}}{2}\]
\[ = \frac{- 1 \pm \sqrt{361}}{2}\]
\[x = \frac{- 1 - 19}{2} or \frac{- 1 + 19}{2}\]
\[ \Rightarrow x = - 10 \text{ or } 9\]
But age cannot be negative so, x = 9.
Thus, Pragati's present age is 9 years.
Solution 2
Let Pragati's present age be x years.
Her age 2 years ago = x - 2 years
Her age 3 years hence = x + 3 years
According to the given condition,
(x - 2) (x + 3) = 84
x(x + 3) -2(x + 3) = 84
x2 + 3x - 2x - 6 = 84
x2 + x - 6 - 84 = 0
x2 + x - 90 = 0
x2 + 10x - 9x - 90 = 0
x(x + 10) -9(x + 10) = 0
(x + 10) (x - 9) = 0
x + 10 = 0 or x - 9 = 0
x = -10 or x = 9
But age cannot be negative
∴ Pragati's present age is 9 years.
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