Given sum of roots = 55 and product of roots = 66.
The quadratic equation is:
x² - (sum of roots)x + (product of roots) = 0
x² - 55x + 66 = 0
Given:
y = 5
Expression: 10y × √(y³ − y²)
Step-by-step:
1. y³ = 5³ = 125
2. y² = 5² = 25
3. y³ − y² = 125 − 25 = 100
4. √(100) = 10
5. 10y = 10 × 5 = 50
6. Final result: 50 × 10 = 500
3x : 2y = 6 : 5
Cross-multiply:
(3x) × 5 = (2y) × 6
15x = 12y
Divide both sides by 3y:
5x/y = 4
x/y = 4/5
So, x : y = 4 : 5
Let son's current age be x and father's current age be y.
x + y = 70 ... (1)
Three years ago,
son's age = x - 3
father's age = y - 3
Given y - 3 = 3(x - 3)
y - 3 = 3x - 9
y = 3x - 6 ... (2)
Putting (2) in (1):
x + 3x - 6 = 70
4x = 76
x = 19
Multiply normally: 1 × 1 = 1
Count decimal places: 2 (from 0.01) + 3 (from 0.001) = 5
Place decimal: 0.00001
Given equation:
3/x + 2/(x+1) = 1
Multiply both sides by x(x+1):
3(x+1) + 2x = x(x+1)
Expanding both sides:
3x + 3 + 2x = x² + x
Combine like terms:
5x + 3 = x² + x
Rearrange the equation:
x² - 4x - 3 = 0
Solve the quadratic equation:
x = (4 ± √(16 + 12)) / 2
x = (4 ± √28) / 2
x = (4 ± 2√7) / 2
x = 2 ± √7
The answer is 2 + √7 or 2 - √7.
x + 1/x = 15
(x + 1/x)² = 15²
x² + 2(x)(1/x) + 1/x² = 225
x² + 2 + 1/x² = 225
x² + 1/x² = 225 - 2
x² + 1/x² = 223
Total of 3 ages = 17 + 16 + 12 = 45
New average = 20, for 4 people → Total = 20 × 4 = 80
Age of 4th person = 80 − 45 = 35
Marked price = Rs 20,000
GST = 12% of 20,000
= 0.12 × 20,000
= Rs 2,400
Total amount = Marked price + GST
= 20,000 + 2,400
= Rs 22,400
To find the number of hours 10 men will take:
First, find the total work hours for 6 men:
6 men * 15 hours = 90 man-hours
Now, divide the total work hours by 10 men:
90 man-hours / 10 men = 9 hours