(1)√x-a/x-b + a/b = √x-b/x-a + b/a, b not equal to a, then value of x is
(2) if x = 2√24/√3+√2 then value of x+√8/x-√8 + x+√12/x-√12 .
(3) if a = 2+√3/2-√3 & b = 2-√3/2+√3 then value of (a square + b square + ab) =
To solve these questions, we will simplify the given expressions and equations step-by-step. Let's solve each question individually:
(1) To find the value of x in the equation (√x-a)/(x-b) + a/b = (√x-b)/(x-a) + b/a, we can follow these steps:
Step 1: Cross-multiply to eliminate the fractions.
(b/a)(√x-a) + a(x-b) = (a/b)(√x-b) + b(x-a)
Step 2: Expand and simplify the equation.
(b/√x)(√x) - (ab/a) + ax - ab = (a/√x)(√x) - (ab/b) + bx - ab
Step 3: Simplify further.
b - ab/√x + ax - ab = a - ab/√x + bx - ab
Step 4: Group like terms.
b - ab/√x + ax - ab - a + ab/√x - bx + ab = 0
Step 5: Simplify.
(ax - bx) + (√x)(b - ab/√x - ab + ab/√x) = 0
Step 6: Cancel out √x terms by multiplying by √x.
x(a - b) + b - a = 0
Step 7: Simplify.
xa - xb + b - a = 0
Step 8: Group like terms.
xa - xb + b - a = 0
Step 9: Simplify further.
xa - xb = a - b
Step 10: Factor out x.
x(a - b) = a - b
Step 11: Divide both sides by (a - b).
x = 1
Therefore, the value of x is 1.
(2) To find the value of x in the expression x + √8/x - √8 + x + √12/x - √12, we can follow these steps:
Step 1: Simplify the expression.
[(x + √8)(x - √8) + (x + √12)(x - √12)] / (x - √8)(x - √12)
Step 2: Expand and simplify.
[(x^2 - (√8)^2) + (x^2 - (√12)^2)] / (x - √8)(x - √12)
Step 3: Simplify further.
[(x^2 - 8) + (x^2 - 12)] / (x - √8)(x - √12)
Step 4: Combine like terms.
(2x^2 - 20) / (x - √8)(x - √12)
Step 5: Factor out 2.
2(x^2 - 10) / (x - √8)(x - √12)
Therefore, the value of x + √8/x - √8 + x + √12/x - √12 is 2(x^2 - 10) / (x - √8)(x - √12).
(3) To find the value of (a^2 + b^2 + ab) given values a = (2 + √3)/(2 - √3) and b = (2 - √3)/(2 + √3), we can follow these steps:
Step 1: Simplify the expression for a.
a = [(2 + √3)(2 + √3)] / [(2 - √3)(2 + √3)]
a = (4 + 4√3 + 3) / (4 - 3)
a = (7 + 4√3)
Step 2: Simplify the expression for b.
b = [(2 - √3)(2 - √3)] / [(2 + √3)(2 - √3)]
b = (4 - 4√3 + 3) / (4 - 3)
b = (7 - 4√3)
Step 3: Find the value of (a^2 + b^2 + ab).
(a^2 + b^2 + ab) = [(7 + 4√3)^2 + (7 - 4√3)^2 + (7 + 4√3)(7 - 4√3)]
Step 4: Expand and simplify.
(a^2 + b^2 + ab) = [49 + 56√3 + 48 + 49 - 56√3 + 48 + (49 - 48)]
(a^2 + b^2 + ab) = 296
Therefore, the value of (a^2 + b^2 + ab) is 296.