Solve the equation 3sqrt(4x-3) - sqrt(5x+1) = 5 for integer values of x.

Step 1: Determine the domain of the variables. For the square roots to be defined, the expressions under them must be non-negative: 4x-3 0 4x 3 x (3)/(4) 5x+1 0 5x -1 x -(1)/(5) Combining these conditions, we must have x (3)/(4). Since x is specified as an integer, x must be 1, 2, 3, . Step 2: Isolate one of the radical terms. The given equation is 3sqrt(4x-3) - sqrt(5x+1) = 5. Move the second radical term to the right side of the equation: 3sqrt(4x-3) = 5 + sqrt(5x+1) Step 3: Square both sides of the equation. (3sqrt(4x-3))^2 = (5 + sqrt(5x+1))^2 9(4x-3) = 5^2 + 2(5)sqrt(5x+1) + (sqrt(5x+1))^2 36x - 27 = 25 + 10sqrt(5x+1) + 5x+1 36x - 27 = 26 + 5x + 10sqrt(5x+1) Step 4: Isolate the remaining radical term. Subtract 26 and 5x from both sides: 36x - 5x - 27 - 26 = 10sqrt(5x+1) 31x - 53 = 10sqrt(5x+1) Step 5: Square both sides again. (31x - 53)^2 = (10sqrt(5x+1))^2 (31x)^2 - 2(31x)(53) + 53^2 = 100(5x+1) 961x^2 - 3286x + 2809 = 500x + 100 Step 6: Rearrange the equation into a standard quadratic form ax^2+bx+c=0. 961x^2 - 3286x - 500x + 2809 - 100 = 0 961x^2 - 3786x + 2709 = 0 Step 7: Solve the quadratic equation for x. Using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -(-3786) ± sqrt((-3786)^2 - 4(961)(2709))2(961) x = 3786 ± sqrt(14333796 - 10413396)1922 x = 3786 ± sqrt(3920400)1922 Calculate the square root: sqrt(3920400) = 1980. x = (3786 ± 1980)/(1922) This gives two possible solutions: x_1 = (3786 + 1980)/(1922) = (5766)/(1922) = 3 x_2 = (3786 - 1980)/(1922) = (1806)/(1922) = (903)/(961) Step 8: Check the solutions against the domain and the original equation. We established that x must be an integer and x (3)/(4). For x_1 = 3: This is an integer and 3 (3)/(4). Substitute x=3 into the original equation: 3sqrt(4(3)-3) - sqrt(5(3)+1) = 3sqrt(12-3) - sqrt(15+1) = 3sqrt(9) - sqrt(16) = 3(3) - 4 = 9 - 4 = 5 This matches the right side of the equation, so x=3 is a valid solution. For x_2 = (903)/(961): This is not an integer, so it is not a valid solution according to the problem's condition that x is an integer. The only integer solution is x=3. The final answer is x=3. Send me the next one 📸