In a quadratic equation with real coefficients, the roots are either both real or come in complex conjugate pairs. If one root is a complex number (for example, \(3 + 5i\)), then the other root must be its complex conjugate, which in this case is \(3 - 5i\).
This property of complex conjugates arises because, when you apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), any complex results from the square root will involve imaginary parts. If the coefficients \(a\), \(b\), and \(c\) are all real numbers, the imaginary part must occur symmetrically for both roots to ensure that their sum and product are also real numbers.
Thus, if you know that \(3 + 5i\) is a root of a quadratic equation with real coefficients, you can confidently conclude that \(3 - 5i\) is also a root of the same equation.