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QUADRATIC EQUATIONS - PROBLEMS ON AGES

**Problem On Ages –**

**Example.1) Five years ago, a father’s age was the square of his daughter’s age. Ten years hence, his age will be twice that of his daughter’s age. Find –
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**(i) The age of the daughter five years ago,
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**(ii) The present age of the father
**

**Ans.) Let the daughter’s age 5 years ago be x years. Then father’s age 5 years ago = x² years
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**Daughter’s present age = (x + 5) years, and father’s present age is (x²+ 5) years
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**Daughter’s age 10 years hence = {(x + 5) + 10} = (x + 15) years
**

**Father’s age 10 years hence = {(x² + 5) + 10} = (x² + 15) years
**

**As per the given condition, ten years hence father’s age will be two times of daughter’s age, then the equation would be –
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** x²+ 15 = 2(x + 15)
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**=> x²+ 15 = 2x + 30
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**=> x²- 2x + 15 – 30 = 0
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**=> x²- 2x – 15 = 0
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**=> x²- (5 – 3)x – 15 = 0
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**=> x²- 5x + 3x – 15 = 0
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**=> x (x – 5) + 3 (x – 5) = 0
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**=> (x – 5) (x + 3) = 0
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**=> (x – 5) = 0 or (x + 3) = 0
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**=> x = 5, or x = - 3
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**=> x = 5 [where x ≠ - 3, since age cannot be negative ]
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**Daughter’s age 5 years ago = 5 years ……………………..(i) (Ans.)
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**Father’s present age (x²+ 5) = (5²+ 5) = 25 + 5 = 30 years………………..(ii) (Ans.)**

**Example.2) The sum of the ages of a father and his son is 45 years. Five years ago, the product of their age was 34. Find the ages of the son and the father.
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**Ans.) Let the present age of the father is x years
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**Then, son’s present age = (45 – x) years
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**Father’s age 5 years ago = (x – 5) years
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**Son’s age 5 years ago = (45 – x – 5) years = (40 – x) years
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**As per the given condition-
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** (x – 5) (40 – x) = 34
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**=> 40x - 200 - x²+ 5x = 34
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**=> 45x - x² = 234
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**=> x²- 45x + 234 = 0
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**=> x²- (39 + 6)x + 234 = 0
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**=> x²- 39x – 6x + 234 = 0
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**=> x (x – 39) – 6 (x – 39) = 0
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**=> (x – 39) (x – 6) = 0
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**=> (x – 39) = 0 or (x – 6) = 0
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**=> x = 39 or x = 6
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**Present age of father cannot be 6. So, we will reject the value x = 6
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**So, x = 39
**

**Hence the present age of father is 39 years and the present age of the son is (45 – x) = (45 – 39) = 6 years. (Ans.)**

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