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 express the given fraction in partial fractions, we first need to factor the denominator. Step 1: Factor the denominator x^2-8x+6. We use the quadratic formula x = -b ± sqrt(b^2-4ac)2a to find the roots of x^2-8x+6=0. Here, a=1, b=-8, c=6. x = -(-8) ± sqrt((-8)^2 - 4(1)(6))2(1) x = 8 ± sqrt(64 - 24)2 x = 8 ± sqrt(40)2 x = 8 ± 2sqrt(10)2 x = 4 ± sqrt(10) So, the denominator can be factored as (x - (4+sqrt(10)))(x - (4-sqrt(10))). Step 2: Set up the partial fraction decomposition. We write the fraction in the form: (2x+1)/(x^2-8x+6) = (A)/(x - (4+sqrt(10))) + (B)/(x - (4-sqrt(10))) Multiply both sides by the common denominator (x - (4+sqrt(10)))(x - (4-sqrt(10))): 2x+1 = A(x - (4-sqrt(10))) + B(x - (4+sqrt(10))) Step 3: Solve for the constants A and B. To find A, substitute x = 4+sqrt(10) into the equation: 2(4+sqrt(10))+1 = A((4+sqrt(10)) - (4-sqrt(10))) + B(0) 8+2sqrt(10)+1 = A(2sqrt(10)) 9+2sqrt(10) = A(2sqrt(10)) A = 9+2sqrt(10)2sqrt(10) = (9+2sqrt(10))sqrt(10)2sqrt(10)sqrt(10) = 9sqrt(10)+2(10)2(10) = 20+9sqrt(10)20 To find B, substitute x = 4-sqrt(10) into the equation: 2(4-sqrt(10))+1 = A(0) + B((4-sqrt(10)) - (4+sqrt(10))) 8-2sqrt(10)+1 = B(-2sqrt(10)) 9-2sqrt(10) = B(-2sqrt(10)) B = 9-2sqrt(10)-2sqrt(10) = (9-2sqrt(10))sqrt(10)-2sqrt(10)sqrt(10) = 9sqrt(10)-2(10)-2(10) = 9sqrt(10)-20-20 = 20-9sqrt(10)20 Step 4: Write the partial fraction decomposition. Substitute the values of A and B back into the partial fraction form: (2x+1)/(x^2-8x+6) = 20+9sqrt(10)20x - (4+sqrt(10)) + 20-9sqrt(10)20x - (4-sqrt(10)) (2x+1)/(x^2-8x+6) = 20+9sqrt(10)20(x - 4 - sqrt(10)) + 20-9sqrt(10)20(x - 4 + sqrt(10)) 3 done, 2 left today. You're making progress.