Question: Find the roots of the following equations:
i. x - \\frac{1}{x}) = 3, x ≠ 0
ii. \\frac{1}{x + 4}) - \\frac{1}{x - 7}) = \\frac{11}{30}), x ≠ -4, 7
Video Explanation
Explanatory Answer
i. x - \\frac{1}{x}) = 3, x ≠ 0
Because x ≠ 0, multiply both sides of the equation by x.
x2 - 1 = 3x
x2 - 3x - 1 = 0
We cannot factorize this equation to find roots. So, let us use the quadratic formula and find the roots of this equation.
The roots of ax2 + bx + c = 0 are \\frac {-b\pm \sqrt {{b}^{2}-4ac}} {2a})
In this equation, a = 1, b = -3 and c = -1
b2 – 4ac = (-3)2 - 4 * 1 * (-1) = 9 + 4 = 13
∴ x = \\frac{-(-3)士 {\sqrt{13}}}{2}) = \\frac{3 士 {\sqrt{13}}}{2})
x = \\frac{3 + {\sqrt{13}}}{2}) and \\frac{3 - {\sqrt{13}}}{2})
ii. \\frac{1}{x + 4}) - \\frac{1}{x - 7}) = \\frac{11}{30}), x ≠ -4, 7
Because x ≠ -4, 7 multiply the equation by (x + 4) (x - 7)
(x - 7) - (x + 4) = \\frac{11}{30})(x + 4) (x - 7)
-11 = \\frac{11}{30})(x + 4) (x - 7)
Multiply the equation by \\frac{30}{11})
-30 = (x + 4) (x - 7)
= x2 - 7x + 4x - 28 + 30 = 0
x2 - 3x + 2 = 0
Method 1: Factorize the quadratic equation to find the values of x
x2 - 2x - x + 2 = x(x - 2) - 1 (x - 2) = 0
(x - 2 )(x - 1) = 0
x = 2 and x = 1
Method 2: Use the quadratic formula to find the values of x
The roots of ax2 + bx + c = 0 are \\frac {-b\pm \sqrt {{b}^{2}-4ac}} {2a})In this equation, a = 1, b = -3 and c = 2
b2 – 4ac = (-3)2 – 4 * 1 * 2 = 9 – 8 = 1 > 0
Therefore, x = -\\frac{-(-3)士{\sqrt1}}{2}) = \\frac{3 士 1}{2})
x= 2 and x = 1