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.
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Fresh day ~LIZZY💞🥺, let's solve. Step 1: Isolate one of the radical terms. We start with the given equation: 3sqrt(4x-3) - sqrt(5x+1) = 5 Add sqrt(5x+1) to both sides to isolate the first radical term: 3sqrt(4x-3) = 5 + sqrt(5x+1) Step 2: Square both sides of the equation. Squaring both sides eliminates the outermost radical on the left and expands the right side using (a+b)^2 = a^2 + 2ab + b^2: (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 Step 3: Simplify the equation and isolate the remaining radical term. Combine like terms on the right side: 36x - 27 = 26 + 5x + 10sqrt(5x+1) Move all non-radical terms to the left side: 36x - 5x - 27 - 26 = 10sqrt(5x+1) 31x - 53 = 10sqrt(5x+1) For this equation to have real solutions, we must have 31x - 53 0, which means x (53)/(31) ≈ 1.71. Step 4: Square both sides again to eliminate the last radical. (31x - 53)^2 = (10sqrt(5x+1))^2 Expand the left side using (a-b)^2 = a^2 - 2ab + b^2: (31x)^2 - 2(31x)(53) + 53^2 = 100(5x+1) 961x^2 - 3286x + 2809 = 500x + 100 Step 5: 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 6: Solve the quadratic equation for x. We use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a = 961, b = -3786, c = 2709. Calculate the discriminant = b^2 - 4ac: = (-3786)^2 - 4(961)(2709) = 14333796 - 10413396 = 3920400 Now find sqrt(): sqrt(3920400) = 1980 Substitute these values into the quadratic formula: x = (-(-3786) ± 1980)/(2(961)) x = (3786 ± 1980)/(1922) This gives two potential solutions: x_1 = (3786 + 1980)/(1922) = (5766)/(1922) = 3 x_2 = (3786 - 1980)/(1922) = (1806)/(1922) = (903)/(961) Step 7: Check the solutions against the domain restrictions and the original equation. The problem states that x must be an integer. From our solutions, x_1 = 3 is an integer, while x_2 = (903)/(961) is not. Also, recall the condition x (53)/(31) ≈ 1.71. x_1 = 3 satisfies this condition. x_2 = (903)/(961) ≈ 0.94, which does not satisfy x 1.71, so it is an extraneous solution. Let's verify x=3 in the original equation: 3sqrt(4(3)-3) - sqrt(5(3)+1) = 5 3sqrt(12-3) - sqrt(15+1) = 5 3sqrt(9) - sqrt(16) = 5 3(3) - 4 = 5 9 - 4 = 5 5 = 5 The solution x=3 is correct. The final answer is 3. Got more? Send 'em.