Quadratic Equations
Roots and Coefficients Relations
Grade 11
Question:
<p>If the roots of the equation \(x^2 - 10cx - 11d = 0\) are \(a, b\) and those of \(x^2 - 10ax - 11b = 0\) are \(c, d\), then find the value of \(a + b + c + d\). (\(a, b, c, d\) are distinct numbers)</p>
Step-by-Step Solution
Key Concept: Set up Vieta's relations for both equations and create a system of equations to solve for the sum.
<p>From the first equation: \(a + b = 10c\) and \(ab = -11d\)</p><p>From the second equation: \(c + d = 10a\) and \(cd = -11b\)</p><p>From \(a + b = 10c\) and \(c + d = 10a\):</p><p>\(a + b = 10c\) ... (1)</p><p>\(c + d = 10a\) ... (2)</p><p>Adding (1) and (2): \(a + b + c + d = 10c + 10a\)</p><p>\(b + d = 9a + 9c\) ... (3)</p><p>Also from \(ab = -11d\) and \(cd = -11b\):</p><p>\(abcd = 121bd\), so \(ac = 121\) (if \(bd \neq 0\))</p><p>Solving the system: \(a + b + c + d = 1210\)</p>
Correct Answer: 1210