SOLVE FOR A VARIABLE INVOLVING SQUARE ROOTS

Problem 1 :

If √1369 + √(0.0615) + x = 37.25, then x is equal to 

(a) 10^-1         (b) 10^-2           (c) 10^-3         (d) None

Solution :

√1369 + √(0.0615) + x = 37.25

37 + √(0.0615) + x = 37.25

√(0.0615) + x = 37.25 - 37

√(0.0615) + x = 0.25

Take square on both sides.

(0.0615) + x = (0.25)²

(0.0615) + x = (25/100)²

(0.0615) + x = (1/4) ⋅ (1/4)

0.0615 + x = 0.0625

x = 0.0625 - 0.0615

x = 0.001

x = 1/1000

x = 10^-3

Problem 2 :

The sum of two numbers is 22 and the sum of their square is 404, then the product of the numbers is 

Solution :

Let the two numbers be x and y.

x + y = 22

x² + y² = 404

(x + y)² = x² + y² + 2xy

Applying the given values, we get

(22)² = 404 + 2xy

484 = 404 + 2xy

2xy = 484 - 404

2xy = 80

xy = 80/2

xy = 40

Problem 3 :

Solution :

Problem 4 :

What is the value of in x in the following :

Solution :

Problem 5 :

If 0.13 ÷ p² = 13, then p equals :

(a) 0.01    (b) 0.1    (c) 10    (d) 100

Solution :

0.13 ÷ p² = 13

p² = 13 ÷ 0.13

p² = 1300/13

p = √100

p = 10

Problem 6 :

√3ⁿ = 729, then the value of n is 

Solution :

√3ⁿ = 729

(3^1/2)ⁿ = 729

3^n/2 = 3⁶

Equating the powers, we get

n/2 = 6

n = 12

Problem 7 :

If √(18 ⋅ 14 ⋅ x) = 84, then x equals.

Solution :

√(18 ⋅ 14 ⋅ x) = 84

√(3 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 7 ⋅ x) = 84

6 √7x = 84

√7x = 84/6

√7x = 14

Take square on both sides.

7x = 196

x = 196/7

x = 28

Problem 8 :

If 28√x + 1426 = 3/4 of 2872, then x equals.

Solution :

28√x + 1426 = 3/4 of 2872

28√x + 1426 = 2154

28√x = 2154 - 1426

28√x = 728

√x = 728/28

√x = 26

x = (26)²

x = 676

Problem 9 :

If √x ÷ √441 = 0.02, then the value of x is :

Solution :

√x ÷ √441 = 0.02

√x ÷ 21 = 0.02

√x  = 0.02 (21)

√x  = 0.42

x = 0.1764

Problem 10 :

If √(x - 1) (y + 2) = 7, x and y being whole numbers, then the values of x and y respectively are

(a)  8, 5   (b)  15, 12    (c)  22, 19    (d)  none

Solution :

√(x - 1) (y + 2) = 7

Applying (a)

x = 8 and y = 5

√(8 - 1) (5 + 2) = 7

√7 ⋅ 7 = 7 (True)

Problem 11 : 

If √(0.04 x 0.4 x a) = 0.004 x 0.4 x √b, then a/b is

Solution :

√(0.04 x 0.4 x a) = 0.004 x 0.4 x √b

Take square on both sides.

0.04 x 0.4 x a = (0.004 x 0.4)² x b

a/b = (0.004 x 0.4)² / (0.04 x 0.4)

a/b = 0.00016

Problem 12 :

Solution :

Problem 13 :

If x = 7 - 4√3, then find the value of (x + 1/x) is

a)  3√3       b)  8√3     c)  14    d)  14 +  8√3

Solution :

x = 7 - 4√3

1/x = 1/(7 - 4√3)

= (7 + 4√3) / (7 - 4√3) (7 + 4√3)

= (7 + 4√3) / (7² - (4√3)²)

= (7 + 4√3) / (49 - (16(3))

= (7 + 4√3) / (49 - 48)

= (7 + 4√3) / 1

x + 1/x = 7 - 4√3 + 7 + 4√3

= 14

So, option c is correct.

Problem 14 :

Th e ratio 125 : x is equivalent to the ratio x : 125. What is the value of x?

Solution :

125 : x = x : 125

125 / x = x / 125

125(125) = x²

x = √(125 x 125)

x = 125

So, the value of x is 125.