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's how to solve the equation: QUESTION 5 We need to solve the equation x + 2sqrt(2x+5) = 8. Step 1: Isolate the radical term. Subtract x from both sides of the equation. 2sqrt(2x+5) = 8 - x For the square root to be defined, 2x+5 0 x -(5)/(2). Also, for the right side to be equal to a non-negative square root, 8-x 0 x 8. So, any valid solution must satisfy -(5)/(2) x 8. Step 2: Square both sides of the equation to eliminate the radical. (2sqrt(2x+5))^2 = (8 - x)^2 4(2x+5) = 64 - 16x + x^2 8x + 20 = 64 - 16x + x^2 Step 3: Rearrange the equation into a standard quadratic form (ax^2 + bx + c = 0). Move all terms to one side. 0 = x^2 - 16x - 8x + 64 - 20 0 = x^2 - 24x + 44 Step 4: Solve the quadratic equation. We can factor the quadratic equation. We need two numbers that multiply to 44 and add up to -24. These numbers are -2 and -22. (x - 2)(x - 22) = 0 This gives two potential solutions: x - 2 = 0 x = 2 x - 22 = 0 x = 22 Step 5: Check for extraneous solutions by substituting each potential solution back into the original equation. Check x = 2: 2 + 2sqrt(2(2)+5) = 8 2 + 2sqrt(4+5) = 8 2 + 2sqrt(9) = 8 2 + 2(3) = 8 2 + 6 = 8 8 = 8 This solution is valid and satisfies the condition -(5)/(2) x 8. Check x = 22: 22 + 2sqrt(2(22)+5) = 8 22 + 2sqrt(44+5) = 8 22 + 2sqrt(49) = 8 22 + 2(7) = 8 22 + 14 = 8 36 = 8 This statement is false, so x=22 is an extraneous solution. It also does not satisfy the condition x 8. The only valid solution is x=2. x=2 Last free one today — make it count tomorrow, or type /upgrade for unlimited.