This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.
ScanSolve AI Answer
Okay Mukisa, I can definitely help you with these radical equations in English. Here are the steps to solve the radical equations: a) Solve sqrt(4-x) - sqrt(6+x) = sqrt(14+2x) Step 1: Determine the domain for x where the terms under the radicals are non-negative. The expressions under the square roots must be non-negative: 4-x 0 x 4 6+x 0 x -6 14+2x 0 2x -14 x -7 Therefore, x must be in the interval [-6, 4]. Step 2: Isolate one radical and then square both sides of the equation. Rearrange the equation to isolate one radical: sqrt(4-x) = sqrt(14+2x) + sqrt(6+x) Square both sides: (sqrt(4-x))^2 = (sqrt(14+2x) + sqrt(6+x))^2 4-x = (14+2x) + (6+x) + 2sqrt((14+2x)(6+x)) 4-x = 20+3x + 2sqrt(84+14x+12x+2x^2) 4-x = 20+3x + 2sqrt(2x^2+26x+84) Step 3: Isolate the remaining radical and simplify. -16-4x = 2sqrt(2x^2+26x+84) Divide both sides by 2: -8-2x = sqrt(2x^2+26x+84) Since the right side is a square root (and thus non-negative), the left side must also be non-negative: -8-2x 0 -2x 8 x -4 So, any solution must satisfy the initial domain conditions and x -4. This means x must be in the interval [-6, -4]. Step 4: Square both sides again and solve the resulting quadratic equation. (-8-2x)^2 = (sqrt(2x^2+26x+84))^2 (2x+8)^2 = 2x^2+26x+84 4x^2+32x+64 = 2x^2+26x+84 2x^2+6x-20 = 0 Divide by 2: x^2+3x-10 = 0 Factor the quadratic equation: (x+5)(x-2) = 0 Thus, x=-5 or x=2. Step 5: Check the solutions against the domain conditions. From Step 3, we required x [-6, -4]. • For x=-5: -5 is within the interval [-6, -4]. • For x=2: 2 is not within the interval [-6, -4]. Thus, x=2 is an extraneous solution. Step 6: Verify the valid solution in the original equation. For x=-5: sqrt(4-(-5)) - sqrt(6+(-5)) = sqrt(14+2(-5)) sqrt(9) - sqrt(1) = sqrt(14-10) 3 - 1 = sqrt(4) 2 = 2 The solution is correct. The solution for equation a) is: x=-5 b) Solve 2sqrt(x-12) + sqrt(2x-3) - 3 = 0 Step 1: Determine the domain for x where the terms under the radicals are non-negative. The expressions under the square roots must be non-negative: x-12 0 x 12 2x-3 0 2x 3 x (3)/(2) Therefore, x must be in the interval [12, ). Step 2: Isolate one radical and then square both sides of the equation. Rearrange the equation: 2sqrt(x-12) + sqrt(2x-3) = 3 Square both sides: (2sqrt(x-12) + sqrt(2x-3))^2 = 3^2 4(x-12) + (2x-3) + 4sqrt((x-12)(2x-3)) = 9 4x-48+2x-3 + 4sqrt(2x^2-3x-24x+36) = 9 6x-51 + 4sqrt(2x^2-27x+36) = 9 Step 3: Isolate the remaining radical and simplify. 4sqrt(2x^2-27x+36) = 9 - (6x-51) 4sqrt(2x^2-27x+36) = 60 - 6x Divide both sides by 2: 2sqrt(2x^2-27x+36) = 30 - 3x Since the left side (square root) is non-negative, the right side must also be non-negative: 30-3x 0 30 3x x 10 Step 4: Compare the conditions for the variable. From Step 1, we required x 12. From Step 3, we required x 10. There is no value of x that can satisfy both conditions (x 12 and x 10) simultaneously. This means there is no valid solution for this equation. The solution for equation b) is: No solution That's 2 down. 3 left today — send the next one.