NCERT Class X Chapter 4: Quadratic Equation Example 1 (i)
NCERT Class X Chapter 4: Quadratic Equation Example 1 (i)
Question:
Represent the following situations mathematically: John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
Given:
Total marbles John and Jivanti initially have = 45Both lost 5 marbles each.
Product of the number of marbles they now have = 124
To Find:
- Represent the following situations mathematically
- Number of marbles each had to start with.
Solution:
Let, John's initial marbles = 'x',
John's initial marbles + Jivanti's initial marbles = 45
Therefore Jivanti's initial marbles = 45 - x
John's initial marbles + Jivanti's initial marbles = 45
Therefore Jivanti's initial marbles = 45 - x
After losing 5 marbles each:
John's marbles = x - 5
Jivanti's marbles = (45 - x) - 5 = 40 - x
John's marbles = x - 5
Jivanti's marbles = (45 - x) - 5 = 40 - x
John's marbles × Jivanti's marbles = 124
⇒ (x - 5)(40 - x) = 124
⇒ 40x - x2 - 200 + 5x = 124
Grouping x terms,
⇒ -x2 + 45x - 200 = 124
Subtract 124 on both sides to make right side of "=" 0,
⇒ -x2 + 45x - 200 - 124 = 0
Grouping constant terms,
⇒ -x2 + 45x - 324 = 0
Multiplying by -1 on both sides,
⇒ x2 - 45x + 324 = 0
⇒ (x - 5)(40 - x) = 124
⇒ 40x - x2 - 200 + 5x = 124
Grouping x terms,
⇒ -x2 + 45x - 200 = 124
Subtract 124 on both sides to make right side of "=" 0,
⇒ -x2 + 45x - 200 - 124 = 0
Grouping constant terms,
⇒ -x2 + 45x - 324 = 0
Multiplying by -1 on both sides,
⇒ x2 - 45x + 324 = 0
The situation is mathematically represented by
x2 - 45x + 324 = 0
x2 - 45x + 324 = 0
WKT,
The quadratic formula, x = -b ± √(b2 - 4ac) 2a
Here, a = 1, b = -45, c = 324
Discriminant (Δ) = b2 - 4ac
⇒ Δ = (-45)2 - 4(1)(324)
⇒ Δ = 2025 - 1296
⇒ Δ = 729
√(Δ) = √(729) = 27
The quadratic formula, x = -b ± √(b2 - 4ac) 2a
Here, a = 1, b = -45, c = 324
Discriminant (Δ) = b2 - 4ac
⇒ Δ = (-45)2 - 4(1)(324)
⇒ Δ = 2025 - 1296
⇒ Δ = 729
√(Δ) = √(729) = 27
Now, substitute values into the quadratic formula:
The quadratic formula, x = (45 ± 27) 2
Two possible values for x:
x1 = (45 + 27) 2 = 72 2 = 36
x2 = (45 - 27) 2 = 18 2 = 9
The quadratic formula, x = (45 ± 27) 2
Two possible values for x:
x1 = (45 + 27) 2 = 72 2 = 36
x2 = (45 - 27) 2 = 18 2 = 9
Case 1: If John initially had 36 marbles.
Jivanti initially had = 45 - 36 = 9 marbles.
After losing 5 marbles each:
John = 36 - 5 = 31 marbles
Jivanti = 9 - 5 = 4 marbles
Product = 31 × 4 = 124 (This matches the given condition)
Jivanti initially had = 45 - 36 = 9 marbles.
After losing 5 marbles each:
John = 36 - 5 = 31 marbles
Jivanti = 9 - 5 = 4 marbles
Product = 31 × 4 = 124 (This matches the given condition)
Case 2: If John initially had 9 marbles.
Jivanti initially had = 45 - 9 = 36 marbles.
After losing 5 marbles each:
John = 9 - 5 = 4 marbles
Jivanti = 36 - 5 = 31 marbles
Product = 4 × 31 = 124 (This also matches the given condition).
Jivanti initially had = 45 - 9 = 36 marbles.
After losing 5 marbles each:
John = 9 - 5 = 4 marbles
Jivanti = 36 - 5 = 31 marbles
Product = 4 × 31 = 124 (This also matches the given condition).
Result:
The number of marbles John and Jivanti had to start with are 9 and 36, or vice-versa.Next Question Solution:
NCERT Class X Chapter 4: Quadratic Equation Example 1 (ii).Explore more in Quadratic Equations chapter:
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