Ch 4: Quadratic Eq (Old Syllabus)
Covers the "Completing the Square" method and advanced word problems from the old NCERT Ex 4.3.
Old Ex 4.3 (Completing the Square Method)
Q1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square.
(i) 2x² - 7x + 3 = 0
Divide the entire equation by 2:
x² - (7/2)x + 3/2 = 0
Move the constant term to the right:
x² - (7/2)x = -3/2
Add the square of half the coefficient of x to both sides. Half of 7/2 is 7/4. Square is 49/16.
x² - (7/2)x + 49/16 = -3/2 + 49/16
Convert LHS into a perfect square and solve RHS:
(x - 7/4)² = (-24 + 49) / 16
(x - 7/4)² = 25/16
Taking square root on both sides:
x - 7/4 = ±5/4
Case 1: x = 7/4 + 5/4 = 12/4 = 3
Case 2: x = 7/4 - 5/4 = 2/4 = 1/2
Roots are 3 and 1/2.
Advanced Word Problems (Old Ex 4.3)
⚠️ Board Tip: The "Train speed" and "Water Taps" questions are 4-mark favorites.
Q8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Solution:
Let the original speed of the train be x km/h.
Time taken = Distance / Speed = 360 / x hours.
New speed = (x + 5) km/h.
New time = 360 / (x + 5) hours.
According to the question, original time - new time = 1 hour:
360 / x - 360 / (x + 5) = 1
Taking 360 common and solving:
360 [ (x + 5 - x) / x(x + 5) ] = 1
360 (5) = x² + 5x
1800 = x² + 5x
x² + 5x - 1800 = 0
Factorizing (45 and 40):
x² + 45x - 40x - 1800 = 0
x(x + 45) - 40(x + 45) = 0
(x - 40)(x + 45) = 0
Speed cannot be negative, so x = 40.
The uniform speed of the train is 40 km/h.
Q9. Two water taps together can fill a tank in 9 ⅜ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Solution:
Let the smaller tap fill the tank in x hours.
Then, larger tap takes (x - 10) hours.
Portion of tank filled by smaller tap in 1 hour = 1/x
Portion of tank filled by larger tap in 1 hour = 1/(x - 10)
Together they fill it in 9 ⅜ = 75/8 hours. So in 1 hour they fill 8/75 of the tank.
1/x + 1/(x - 10) = 8/75
Taking LCM:
(x - 10 + x) / x(x - 10) = 8/75
(2x - 10) / (x² - 10x) = 8/75
75(2x - 10) = 8(x² - 10x)
150x - 750 = 8x² - 80x
8x² - 230x + 750 = 0
Dividing by 2: 4x² - 115x + 375 = 0
Using the quadratic formula to solve this yields x = 25 or x = 3.75.
If x = 3.75, the larger tap takes 3.75 - 10 = negative time (not possible). So x = 25.
Smaller tap takes 25 hours, Larger tap takes 15 hours.