Rational Equations Solve for X and check.?

A) (x+10)/(x^2-2)=(4/x)

cross-multiply...

x(x + 10) = 4(x² - 2)
x² + 10x = 4x² - 8
3x² - 10x - 8 = 3x² - 12x + 2x - 8 = 3x(x - 4) + 2(x - 4) = (x - 4)(3x + 2) = 0

x = 4 or x = - 2/3

check...do something !

one at a time is enough for anyone !!

A)
(x + 10)/(x² - 2) = 4/x
4(x² - 2) = x(x + 10)
4x² - 8 = x² + 10x
3x² - 10x - 8 = 0
(x - 4)(3x + 2) = 0
x = 4 or x = -2/3

Check :
When x = 4 :
L.H.S. = (4 + 10)/(4² - 2) = 1
R.H.S. = 4/4 = 1
As L.H.S. = R.H.S.
The root x = 4 is accepted.

When x = -2/3 :
L.H.S. = [(-2/3) + 10]/[(-2/3)² - 2)] = -6
R.H.S. = 4/(-2/3) = -6
As L.H.S. = R.H.S.
The root x = -2/3 is accepted.

Ans : x = 4 or x = -2/3


B)
x / 5 = [x / (x - 5)] - 1
x(x - 5) / 5(x - 5) = [5x / 5(x - 5)] - [5(x - 5) / 5(x - 5)]

When x ≠ 5 :
x(x - 5) = 5x - 5(x - 5)
x² - 5x = 5x - 5x + 25
x² - 5x - 25 = 0
x = (5 + 5√5)/2 or x = (5 - 5√5)/2

Check :
When x = (5 + 5√5)/2 :
L.H.S. = [(5 + 5√5)/2] / 5 = (1 + √5)/2
R.H.S. = [(5 + 5√5)/2] / {[(5 + 5√5)/2] - 5)} - 1 = (1 + √5)/2
As L.H.S. = R.H.S.
The root x = (5 + 5√5)/2 is accepted.

When x = (5 - 5√5)/2 :
L.H.S. = [(5 - 5√5)/2] / 5 = (1 - √5)/2
R.H.S. = [(5 - 5√5)/2] / {[(5 - 5√5)/2] - 5)} - 1 = (1 - √5)/2
As L.H.S. = R.H.S.
The root x = (5 - 5√5)/2 is accepted.

Ans: x = (5 + 5√5)/2 or x = (5 - 5√5)/2

(x + 10)/(x² - 2) = 4/x → where x ≠ ± √2 and where x ≠ 0

(x + 10)/(x² - 2) = 4/x → you make the cross-multiply

x.(x + 10) = 4.(x² - 2) → you expand

x² + 10x = 4x² - 8

x² + 10x - 4x² + 8 = 0

- 3x² + 10x + 8 = 0

- 3.[x² - (10/3).x - (8/3)] = 0

x² - (10/3).x - (8/3) = 0

x² - (10/3).x + (5/3)² - (5/3)² - (8/3) = 0

x² - (10/3).x + (5/3)² - (25/9) - (24/9) = 0

x² - (10/3).x + (5/3)² - (49/9) = 0

x² - (10/3).x + (5/3)² = 49/9

[x - (5/3)]² = (± 7/3)²

x - (5/3) = ± 7/3

x = (5/3) ± (7/3)

x = (5 ± 7)/3


First case: x = (5 + 7)/3 → x = 12/3 → x = 4

Second case: x = (5 - 7)/3 → x = - 2/3


→ Solution = { - 2/3 ; 4 }




x/5 = [x/(x - 5)] - 1 → where x ≠ 5

x/5 = [x/(x - 5)] - 1 → you multiply by 5 both sides

x = [5x/(x - 5)] - 5

x = [5x - 5.(x - 5)] / (x - 5) → you make the cross-multiply

x.(x - 5) = 5x - 5.(x - 5) → you expand

x² - 5x = 5x - 5x + 5

x² - 5x = 5

x² - 5x + (5/2)² - (5/2)² = 5

x² - 5x + (5/2)² = 5 + (5/2)²

x² - 5x + (5/2)² = (20/4) + (25/4)

x² - 5x + (5/2)² = 45/4

[x - (5/2)]² = [± √(45/4)]²

x - (5/2) = ± √(45/4)

x - (5/2) = ± (3√5)/2

x = (5/2) ± [(3√5)/2]

x = (5 ± 3√5)/2


First case: x = (5 + 3√5)/2

Second case: x = (5 - 3√5)/2


→ Solution = { (5 - 3√5)/2 ; (5 + 3√5)/2 }