f5 quadratic equation

Let x be the number

1. The square root of a positive number is 20 less than the original number.find the number
√x = x – 20
x - √x – 20 = 0
(√x)^2 -√x – 20 = 0
(√x – 5) (√x + 4) = 0
√x = 5
x = 25
√x = - 4. Ignore this

x = 25 (Answer)

2.When 3 is added to a given number,the result is ten times the reciprocal of the number.find the possible value of the number
3 + x = 10(1/x)
3x + x^2 = 10
x^2 + 3x – 10 = 0
(x + 5) (x -2) = 0

x = 2 or x = - 5 (Answer)

3.The sum of a number and its positive square root is 12.find the number
x + √x = 12
(√x)^2 + √x – 12 = 0
(√x + 4)( √x – 3) = 0
√x = -4
x = 16
√x = 3
x = 9

x= 16 or x = 9 (Answer)

4. The sum of a number and its reciprocal is 41/20.find all the possible numbers
x + 1/x = 41/20
x^2 + 1 = 41x/20
20x^2 + 20 = 41x
20x^2– 41x + 20 = 0
Using quadratic formula
x = [-(-41)+- √ (-41)^2 – 4(20)(20))]/2(20)
x = [41+- √ (1681 – 1600)]/40
x = [41+- √ (81)]/40
x = [41+ 9]/40 or x = [41- 9]/40
x = 50/40 = 1.25 or x = 32/40 = 0.8

x = 1.25 or x = 0.8 (Answer)

8. Mr. Chan buys some flour with $80. The price of flour is $x per kg. If the price is increased by $2 per kg, he will buy 2 kg less than with $80. Find the value of x.(Ans:8)

10. Water from pipe A can fill up a swimming pool in x hours. Water from pipe B takes two
hours to fill up the same swimming pool. If the two pipes are used together, three quarters of the swimming pool can be filled up in one hour. Find the possible value(s) of x. (Hint: Let the capacity of the swimming pool be S cm3.)(Ans:2)

11.In a two-digit number, the tens digit is greater than the units digit by 1. If the sum of the square of the tens digit and the units digit is 41. Find the number.(Ans:54)
.In a two-digit number, the tens digit is greater than the units digit by 1. If the sum of the square of the tens digit and the units digit is 41. Find the number.(Ans:54)

1.
√x = x - 20
x = (x - 20)²
x = x² - 40x + 400
x² - 41x + 400 = 0
(x - 25)(x - 16) = 0
x = 25 or x = 16 (rejected)

3.
x + √x = 12
√x = 12 - x
x = (12 - x)²
x = 144 - 24x + x²
x² - 25x + 144 = 0
(x - 9)(x - 16) = 0
x = 9 or x = 16