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
Hey Faith, good to see you again. Step 1: Rearrange the given equation into the standard quadratic form ax^2 + bx + c = 0. The given equation is x^2 - 3x + 1 = p(x-3). Expand the right side: x^2 - 3x + 1 = px - 3p Move all terms to the left side: x^2 - 3x - px + 1 + 3p = 0 Factor out x: x^2 + (-3 - p)x + (1 + 3p) = 0 Step 2: Identify the coefficients a, b, and c. From the standard quadratic form ax^2 + bx + c = 0, we have: a = 1 b = -3 - p c = 1 + 3p Step 3: Apply the condition for equal roots. For an equation to have equal roots, the discriminant () must be zero. The discriminant is given by the formula = b^2 - 4ac. Set the discriminant to zero: b^2 - 4ac = 0 Substitute the values of a, b, and c: (-3 - p)^2 - 4(1)(1 + 3p) = 0 Step 4: Solve the equation for p. Simplify the equation: (p + 3)^2 - 4(1 + 3p) = 0 p^2 + 6p + 9 - 4 - 12p = 0 Combine like terms: p^2 - 6p + 5 = 0 Factor the quadratic equation: (p - 1)(p - 5) = 0 Set each factor to zero to find the possible values of p: p - 1 = 0 p = 1 p - 5 = 0 p = 5 The possible values of p are 1, 5. Send me the next one 📸