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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Here are the solutions to the radical equations: 22. Solve sqrt(2 - x) = x - 2 Step 1: Determine the domain restrictions. For the square root sqrt(2-x) to be defined, 2-x 0 x 2. For the right side x-2 to be non-negative (as it equals a square root), x-2 0 x 2. Combining these, the only possible value for x is x=2. Step 2: Check if x=2 is a solution. Substitute x=2 into the equation: sqrt(2 - 2) = 2 - 2 sqrt(0) = 0 0 = 0 This is true, so x=2 is a valid solution. Alternatively, by squaring both sides: Step 1: Square both sides of the equation. (sqrt(2 - x))^2 = (x - 2)^2 2 - x = x^2 - 4x + 4 Step 2: Rearrange the equation into a standard quadratic form. 0 = x^2 - 4x + x + 4 - 2 0 = x^2 - 3x + 2 Step 3: Solve the quadratic equation by factoring. (x - 1)(x - 2) = 0 This gives two potential solutions: x = 1 or x = 2. Step 4: Check for extraneous solutions using the original equation. For x=1: sqrt(2 - 1) = 1 - 2 sqrt(1) = -1 1 = -1 This is false, so x=1 is an extraneous solution. For x=2: sqrt(2 - 2) = 2 - 2 sqrt(0) = 0 0 = 0 This is true, so x=2 is a valid solution. The final answer is x = 2. --- 23. Solve sqrt(43 - x) - x + 1 = 0 Step 1: Isolate the radical term. sqrt(43 - x) = x - 1 Step 2: Determine the domain restrictions. For the square root sqrt(43-x) to be defined, 43-x 0 x 43. For the right side x-1 to be non-negative, x-1 0 x 1. So, any valid solution must satisfy 1 x 43. Step 3: Square both sides of the equation. (sqrt(43 - x))^2 = (x - 1)^2 43 - x = x^2 - 2x + 1 Step 4: Rearrange the equation into a standard quadratic form. 0 = x^2 - 2x + x + 1 - 43 0 = x^2 - x - 42 Step 5: Solve the quadratic equation by factoring. We look for two numbers that multiply to -42 and add to -1. These are -7 and 6. (x - 7)(x + 6) = 0 This gives two potential solutions: x = 7 or x = -6. Step 6: Check for extraneous solutions using the domain restrictions or the original equation. For x=7: This satisfies 1 7 43. Substitute into the original equation: sqrt(43 - 7) - 7 + 1 = 0 sqrt(36) - 6 = 0 6 - 6 = 0 0 = 0 This is true, so x=7 is a valid solution. For x=-6: This does not satisfy x 1 (since -6 < 1). So x=-6 is an extraneous solution. Alternatively, substitute into the original equation: sqrt(43 - (-6)) - (-6) + 1 = 0 sqrt(49) + 6 + 1 = 0 7 + 6 + 1 = 0 14 = 0 This is false, so x=-6 is an extraneous solution. The final answer is x = 7. --- 24. Solve 2sqrt(x + 6) + 2 = x Step 1: Isolate the radical term. 2sqrt(x + 6) = x - 2 Step 2: Determine the domain restrictions. For the square root sqrt(x+6) to be defined, x+6 0 x -6. For the right side x-2 to be non-negative (as it equals twice a square root), x-2 0 x 2. So, any valid solution must satisfy x 2. Step 3: Square both sides of the equation. (2sqrt(x + 6))^2 = (x - 2)^2 4(x + 6) = x^2 - 4x + 4 4x + 24 = x^2 - 4x + 4 Step 4: Rearrange the equation into a standard quadratic form. 0 = x^2 - 4x - 4x + 4 - 24 0 = x^2 - 8x - 20 Step 5: Solve the quadratic equation by factoring. We look for two numbers that multiply to -20 and add to -8. These are -10 and 2. (x - 10)(x + 2) = 0 This gives two potential solutions: x = 10 or x = -2. Step 6: Check for extraneous solutions using the domain restrictions or the original equation. For x=10: This satisfies x 2 (since 10 2). Substitute into the original equation: 2sqrt(10 + 6) + 2 = 10 2sqrt(16) + 2 = 10 2(4) + 2 = 10 8 + 2 = 10 10 = 10 This is true, so x=10 is a valid solution. For x=-2: This does not satisfy x 2 (since -2 < 2). So x=-2 is an extraneous solution. Alternatively, substitute into the original equation: 2sqrt(-2 + 6) + 2 = -2 2sqrt(4) + 2 = -2 2(2) + 2 = -2 4 + 2 = -2 6 = -2 This is false, so x=-2 is an extraneous solution. The final answer is x = 10.