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.
ScanSolve AI Answer
You're on a roll — let's solve this equation. Step 1: Expand both sides of the equation. The given equation is (11-3x)^2 = 4(x+4)(x-1). Expand the left side (11-3x)^2 using the formula (a-b)^2 = a^2 - 2ab + b^2: (11-3x)^2 = 11^2 - 2(11)(3x) + (3x)^2 = 121 - 66x + 9x^2 Expand the right side 4(x+4)(x-1). First, multiply the binomials (x+4)(x-1): (x+4)(x-1) = x(x-1) + 4(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4 Now, multiply by 4: 4(x^2 + 3x - 4) = 4x^2 + 12x - 16 Step 2: Set the expanded left side equal to the expanded right side. 121 - 66x + 9x^2 = 4x^2 + 12x - 16 Step 3: Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0. Subtract 4x^2, 12x, and -16 from both sides to move all terms to one side: 9x^2 - 4x^2 - 66x - 12x + 121 + 16 = 0 5x^2 - 78x + 137 = 0 Step 4: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=5, b=-78, and c=137. Calculate the discriminant = b^2 - 4ac: = (-78)^2 - 4(5)(137) = 6084 - 20(137) = 6084 - 2740 = 3344 Now, substitute the values into the quadratic formula: x = -(-78) ± sqrt(3344)2(5) x = 78 ± sqrt(3344)10 Step 5: Simplify the square root. Factorize 3344 to find any perfect square factors: 3344 = 16 × 209 So, sqrt(3344) = sqrt(16 × 209) = sqrt(16) × sqrt(209) = 4sqrt(209). Substitute this back into the expression for x: x = 78 ± 4sqrt(209)10 Step 6: Simplify the fraction by dividing the numerator and denominator by 2. x = 2(39 ± 2sqrt(209))10 x = 39 ± 2sqrt(209)5 The solutions for x are: x_1 = 39 + 2sqrt(209)5 x_2 = 39 - 2sqrt(209)5 The solutions are x = 39 ± 2sqrt(209)5. Got more? Send 'em.