This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve the equation x - 6 + (2)/(x) = 0, with the condition x ≠ 0: Step 1: Multiply the entire equation by x to eliminate the fraction. Since x ≠ 0, we can multiply both sides by x without changing the validity of the equation. x · (x) - 6 · (x) + (2)/(x) · (x) = 0 · (x) x^2 - 6x + 2 = 0 Step 2: Use the quadratic formula to solve for x. The equation is in the standard quadratic form ax^2 + bx + c = 0, where a=1, b=-6, and c=2. The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. Substitute the values of a, b, and c into the formula: x = -(-6) ± sqrt((-6)^2 - 4(1)(2))2(1) x = 6 ± sqrt(36 - 8)2 x = 6 ± sqrt(28)2 Step 3: Simplify the radical and the expression for x. Simplify sqrt(28): sqrt(28) = sqrt(4 × 7) = sqrt(4) × sqrt(7) = 2sqrt(7) Substitute this back into the expression for x: x = 6 ± 2sqrt(7)2 Factor out 2 from the numerator: x = 2(3 ± sqrt(7))2 Cancel out the common factor of 2: x = 3 ± sqrt(7) Step 4: Verify the solutions against the condition x ≠ 0. The two solutions are x_1 = 3 + sqrt(7) and x_2 = 3 - sqrt(7). Since sqrt(7) is approximately 2.646, neither 3 + sqrt(7) nor 3 - sqrt(7) is equal to zero. Both solutions are valid. The final answers are x = 3 + sqrt(7), 3 - sqrt(7).