One solution of a quadratic equation is 3+5i

Question

b) Explain how you know.

Answer 1

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.