how to calculate

Question

Solve for x if: 1. x+y+yj+2j=8j+7 2. 2^(5x-2) = 5^(3x+1) 3. 3.3^x = 28 4. 3^(5x+1) = 5^(2x+2) 5. (X+1)(x+4) = -8 6. 3(sec^2) x - 5=4tan x ; 0

Answer 1

1.

x+y+yj+2j = 8j+7

x + (y + 1)j + 2j = 8j + 7                          (Using distributive property)

Subtract 2j from each side.

x + (y + 1)j + 2j - 2j = 8j + 7 - 2j

x + (y + 1)j = 6j + 7

Subtract (y + 1)j from each side.

x + (y + 1)j - (y + 1)j = 6j + 7 - (y + 1)j

x = (6 - (y + 1)) j + 7

Answer 2

2)

2^(5x-2) = 5^(3x+1)

Take logarithms both sides.

log [2^(5x-2)] = log [5^(3x+1)]

(5x - 2) log 2 =(3x + 1) log 5

5x log 2 - 2log 2 = 3xlog 5 + log 5

5x log 2 - 3xlog 5= 2log 2 + log 5

x (5 log 2 - 3log 5) = 2log 2 + log 5

x = (2log 2 + log 5)/(5 log 2 - 3log 5)

Answer 3

3)

3.3^x = 28

Using the product of exponents, a^m * a^n = a^ (m+n)

3^ (1+x) =28

Take logarithm both sides.

log [3^ (1+x)]  = log 28

(1+x) log 3 = log 28

1+x = log 28 / log 3

x = [log 28 / log 3] -1

   = 2.033

Answer 4

4.

3^(5x+1) = 5^(2x+2)

Take logarithms both sides.

log [3^(5x+1)] = log [5^(2x+2)]

(5x + 1) log 3 =(2x + 2) log 5

5x log 3 - log 3 = 2xlog 5 + 2log 5

5x log 3 - 2xlog 5= log 3 + 2 log 5

x (5 log 3 - 2log 5) = log 3 + 2log 5

x = (log 3 + 2 log 5)/(5 log 3 - 2log 5)

Answer 5

5)

(x+1)(x+4) = -8

x + 1 = - 8      or    x + 4  = - 8

x + 1 = - 8

x + 1-1 =-8-1 

x = -9

x + 4  = - 8

x + 4 - 4  = - 8 - 4

x = -12

Answer 6

6)

3sec² x -  5=4tan x

sec² x can be written as tan² x + 1.

3(tan² x + 1) - 5=4tan x

3tan² x +3 -5 = 4tan x

3tan² x - 4tan x - 2 = 0

Solve the quadratic equation to get,

tan x = 1.72      or     tan x = -0.387

tan x = 1.72

tan x = tan 59.83

The general solution of tan(θ) = tan(α) is θ = nπ + α, where n is an integer.

x = nπ + 59.83

n = 0, x = (0)π + 59.83 = 59.83.

n = 1, x = (1)π + 59.83 = 239.83.

tan x = - 0.387

tan x = tan (- 21.16 )

The general solution of tan(θ) = tan(α) is θ = nπ + α, where n is an integer.

x = nπ - 21.16

n = 1, x = (1)π - 21.16 = 158.84..

n = 2, x = (2)π - 21.16 = 338.84.

Therefore, the solutions of the given equation are x = 59.83, 158.84, 239.83, and 338.84 in the interval [0, 2π).