Solve sqrt(14-x) = sqrt(x-4) + sqrt(x-1)

Here are the solutions for the given radical equations. (v) Solve sqrt(14-x) = sqrt(x-4) + sqrt(x-1) Step 1: Determine the domain of the equation. For the square roots to be defined, we must have: 14-x 0 x 14 x-4 0 x 4 x-1 0 x 1 Combining these conditions, the domain is 4 x 14. Step 2: Square both sides of the equation. (sqrt(14-x))^2 = (sqrt(x-4) + sqrt(x-1))^2 14-x = (x-4) + (x-1) + 2sqrt((x-4)(x-1)) 14-x = 2x-5 + 2sqrt(x^2-5x+4) Step 3: Isolate the remaining square root term. 14-x - (2x-5) = 2sqrt(x^2-5x+4) 19-3x = 2sqrt(x^2-5x+4) For this equation to hold, we must have 19-3x 0, which means 3x 19, or x (19)/(3) ≈ 6.33. Step 4: Square both sides again. (19-3x)^2 = (2sqrt(x^2-5x+4))^2 361 - 114x + 9x^2 = 4(x^2-5x+4) 361 - 114x + 9x^2 = 4x^2 - 20x + 16 Step 5: Rearrange into a quadratic equation. 9x^2 - 4x^2 - 114x + 20x + 361 - 16 = 0 5x^2 - 94x + 345 = 0 Step 6: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2-4ac)2a. x = -(-94) ± sqrt((-94)^2 - 4(5)(345))2(5) x = 94 ± sqrt(8836 - 6900)10 x = 94 ± sqrt(1936)10 x = (94 ± 44)/(10) The two potential solutions are: x_1 = (94 + 44)/(10) = (138)/(10) = 13.8 x_2 = (94 - 44)/(10) = (50)/(10) = 5 Step 7: Check the solutions against the domain and the condition x (19)/(3). For x_1 = 13.8: This value satisfies 4 x 14, but it does not satisfy x (19)/(3) (since 13.8 > 6.33). Thus, x=13.8 is an extraneous solution. For x_2 = 5: This value satisfies 4 x 14 and x (19)/(3). Substitute x=5 into the original equation: sqrt(14-5) = sqrt(5-4) + sqrt(5-1) sqrt(9) = sqrt(1) + sqrt(4) 3 = 1 + 2 3 = 3 This is true. The solution is x=5. (vi) Solve sqrt(4x+1) + sqrt(2x-3) = 4 Step 1: Determine the domain of the equation. For the square roots to be defined, we must have: 4x+1 0 4x -1 x -(1)/(4) 2x-3 0 2x 3 x (3)/(2) Combining these conditions, the domain is x (3)/(2). Step 2: Square both sides of the equation. (sqrt(4x+1) + sqrt(2x-3))^2 = 4^2 (4x+1) + (2x-3) + 2sqrt((4x+1)(2x-3)) = 16 6x-2 + 2sqrt(8x^2-12x+2x-3) = 16 6x-2 + 2sqrt(8x^2-10x-3) = 16 Step 3: Isolate the remaining square root term. 2sqrt(8x^2-10x-3) = 16 - (6x-2) 2sqrt(8x^2-10x-3) = 18 - 6x Divide by 2: sqrt(8x^2-10x-3) = 9 - 3x For this equation to hold, we must have 9-3x 0, which means 3x 9, or x 3. Step 4: Square both sides again. (sqrt(8x^2-10x-3))^2 = (9-3x)^2 8x^2-10x-3 = 81 - 54x + 9x^2 Step 5: Rearrange into a quadratic equation. 0 = 9x^2 - 8x^2 - 54x + 10x + 81 + 3 0 = x^2 - 44x + 84 Step 6: Solve the quadratic equation by factoring. We need two numbers that multiply to 84 and add to -44. These numbers are -2 and -42. (x-2)(x-42) = 0 The two potential solutions are: x_1 = 2 x_2 = 42 Step 7: Check the solutions against the domain and the condition x 3. For x_1 = 2: This value satisfies x (3)/(2) and x 3. Substitute x=2 into the original equation: sqrt(4(2)+1) + sqrt(2(2)-3) = 4 sqrt(8+1) + sqrt(4-3) = 4 sqrt(9) + sqrt(1) = 4 3 + 1 = 4 4 = 4 This is true. For x_2 = 42: This value satisfies x (3)/(2), but it does not satisfy x 3 (since 42 > 3). Thus, x=42 is an extraneous solution. The solution is $x=