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
To find the values of p and q, we will use the properties of roots of a quadratic equation with real coefficients. Step 1: Identify the roots of the equation. Given that 2 + isqrt(3) is a root of the equation x^2 + px + q = 0, and p, q R (real coefficients), the complex conjugate of this root must also be a root. So, if x_1 = 2 + isqrt(3), then the second root is x_2 = 2 - isqrt(3). Step 2: Use Vieta's formulas for the sum of the roots. For a quadratic equation x^2 + px + q = 0, the sum of the roots is given by x_1 + x_2 = -p. (2 + isqrt(3)) + (2 - isqrt(3)) = -p 2 + 2 + isqrt(3) - isqrt(3) = -p 4 = -p p = -4 Step 3: Use Vieta's formulas for the product of the roots. For a quadratic equation x^2 + px + q = 0, the product of the roots is given by x_1 · x_2 = q. (2 + isqrt(3))(2 - isqrt(3)) = q This is in the form (a+b)(a-b) = a^2 - b^2. 2^2 - (isqrt(3))^2 = q 4 - (i^2 · (sqrt(3))^2) = q Since i^2 = -1 and (sqrt(3))^2 = 3: 4 - (-1 · 3) = q 4 - (-3) = q 4 + 3 = q q = 7 Step 4: State the ordered pair (p, q). From the calculations, p = -4 and q = 7. Therefore, (p, q) = (-4, 7). Comparing this with the given options, the correct option is (A). The final answer is (A) (-4, 7). Send me the next one 📸