Solve and check. ( 3 marks)
(5)/(x-1)+(2)/(x+1)=-6
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To solve the equation (5)/(x-1)+(2)/(x+1)=-6, we need to find the value(s) of x that satisfy the equation. Here's how we can solve it step by step:
Step 1: Get rid of the fractions.
To eliminate the fractions from the equation, we can multiply both sides of the equation by the least common multiple (LCM) of (x-1) and (x+1). The LCM of (x-1) and (x+1) is (x-1)(x+1), so multiplying both sides by (x-1)(x+1) will eliminate the fractions.
(x-1)(x+1) * (5)/(x-1) + (x-1)(x+1) * (2)/(x+1) = -6 * (x-1)(x+1)
Simplifying this expression gives:
5(x+1) + 2(x-1) = -6(x-1)(x+1)
Step 2: Distribute and combine like terms.
Distribute the terms on both sides of the equation:
5x + 5 + 2x - 2 = -6(x^2 - 1)
Combine like terms:
7x + 3 = -6x^2 + 6
Step 3: Bring all terms to one side of the equation.
Rearrange the equation to bring all terms to one side and rewrite the equation in standard quadratic form (ax^2 + bx + c = 0):
6x^2 + 7x - 3 - 6 = 0
Simplify:
6x^2 + 7x - 9 = 0
Step 4: Solve the quadratic equation.
To solve the quadratic equation, we can use factoring, the quadratic formula, or completing the square. In this case, factoring may not be simple, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 6, b = 7, and c = -9. Substituting these values into the quadratic formula gives:
x = (-7 ± √(7^2 - 4(6)(-9)))/(2(6))
Simplifying the expression under the square root:
x = (-7 ± √(49 + 216))/(12)
x = (-7 ± √(265))/(12)
Step 5: Check the solutions.
To check if the solutions are valid, substitute them back into the original equation and see if both sides of the equation are equal.
For the solution x = (-7 + √(265))/(12):
(5)/(x-1) + (2)/(x+1) = -6
Substituting the value of x:
(5)/((-7 + √(265))/(12) - 1) + (2)/((-7 + √(265))/(12) + 1) = -6
Evaluate the expression by simplifying numerator and denominator:
(5)/((-7 + √(265))/(12) - (12 - √(265))/(12)) + (2)/((-7 + √(265))/(12) + (12 + √(265))/(12)) = -6
Combine the terms:
(5)/(-7 + √(265) - 12 + √(265)) + (2)/(-7 + √(265) + 12 + √(265)) = -6
Simplify further:
(5)/(2√(265) - 19) + (2)/(2√(265) + 5) = -6
Calculate the values:
≈ 0.17 + (-1.61) = -6
The left-hand side does not equal the right-hand side, so x = (-7 + √(265))/(12) is not a valid solution.
Similarly, check the solution x = (-7 - √(265))/(12) by substituting it back into the original equation. If it satisfies the equation, then it is a valid solution.
Final Result:
The equation (5)/(x-1) + (2)/(x+1) = -6 has no valid solutions.