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
Here's how to form the quadratic equation: 22) Form a quadratic equation where roots are (3)/(4) and (1)/(3) giving your answer in the form ax^2 + bx + c = 0, whereas a, b and c are integers. Step 1: Identify the given roots. The roots are x_1 = (3)/(4) and x_2 = (1)/(3). Step 2: Write the factors corresponding to these roots. If x = (3)/(4), then x - (3)/(4) = 0. If x = (1)/(3), then x - (1)/(3) = 0. Step 3: Form the quadratic equation by multiplying the factors. (x - (3)/(4))(x - (1)/(3)) = 0 Expand the product: x(x - (1)/(3)) - (3)/(4)(x - (1)/(3)) = 0 x^2 - (1)/(3)x - (3)/(4)x + (-(3)/(4))(-(1)/(3)) = 0 x^2 - (1)/(3)x - (3)/(4)x + (3)/(12) = 0 x^2 - (1)/(3)x - (3)/(4)x + (1)/(4) = 0 Step 4: Combine the x terms and clear the denominators to get integer coefficients. Find a common denominator for the fractions (1)/(3) and (3)/(4), which is 12. x^2 - (4)/(12)x - (9)/(12)x + (1)/(4) = 0 x^2 - (13)/(12)x + (1)/(4) = 0 To eliminate the fractions, multiply the entire equation by the least common multiple of the denominators (12 and 4), which is 12. 12(x^2 - (13)/(12)x + (1)/(4)) = 12(0) 12x^2 - 12((13)/(12))x + 12((1)/(4)) = 0 12x^2 - 13x + 3 = 0 3 done, 2 left today. You're making progress.