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
Step 1: Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0. The given equation is: 3x^2 + 2x + 1 = 12x + 21 Subtract 12x from both sides: 3x^2 + 2x - 12x + 1 = 21 3x^2 - 10x + 1 = 21 Subtract 21 from both sides: 3x^2 - 10x + 1 - 21 = 0 3x^2 - 10x - 20 = 0 Step 2: Solve the quadratic equation using the quadratic formula. The quadratic equation is in the form ax^2 + bx + c = 0, where a=3, b=-10, and c=-20. The quadratic formula is: x = -b ± sqrt(b^2 - 4ac)2a Substitute the values of a, b, and c into the formula: x = -(-10) ± sqrt((-10)^2 - 4(3)(-20))2(3) x = 10 ± sqrt(100 - (-240))6 x = 10 ± sqrt(100 + 240)6 x = 10 ± sqrt(340)6 Step 3: Simplify the square root and the expression for x. Simplify sqrt(340): sqrt(340) = sqrt(4 × 85) = sqrt(4) × sqrt(85) = 2sqrt(85) Substitute this back into the expression for x: x = 10 ± 2sqrt(85)6 Factor out 2 from the numerator: x = 2(5 ± sqrt(85))6 Divide the numerator and denominator by 2: x = 5 ± sqrt(85)3 The two solutions for x are: x_1 = 5 + sqrt(85)3 x_2 = 5 - sqrt(85)3 The final answer is x = 5 ± sqrt(85)3.