Question

\(4-i\) is a solution of a quadratic equation with real coefficients. Find the other solution.

Short answer

The other solution is \(4+i\).

Step-by-step solution

Identify properties of the solutions

Since the quadratic equation has real coefficients and one of the solutions is a complex number, the other solution must be the complex conjugate of the given solution. This means if the solution is given as \(4-i\), the complex conjugate is \(4+i\).

Write the other solution

Simply write the complex conjugate of \(4-i\). The complex conjugate of a complex number \(a+bi\) is \(a-bi\).

Validate the result

Verify that the given solution and the found solution are complex conjugates. Here, \(4-i\) and \(4+i\) are indeed complex conjugates, confirming the correctness.

Key concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the form \[ax^2 + bx + c = 0\], where *a*, *b*, and *c* are constants, and *x* represents the variable. Quadratic equations always have two solutions. These solutions could be real or complex numbers. The solutions are found using the quadratic formula which is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. The part under the square root, called the discriminant (\[b^2 - 4ac\]), helps determine the nature of the roots:

• If \[b^2 - 4ac > 0\], the quadratic equation has two distinct real solutions.
• If \[b^2 - 4ac = 0\], the quadratic equation has one real solution (a repeated root).
• If \[b^2 - 4ac < 0\], the quadratic equation has two complex solutions.

When working with equations with real coefficients, if one root is complex, the other root must also be complex and specifically, it must be the complex conjugate of the first root. This ensures the coefficients of the quadratic equation remain real.

complex conjugate

The complex conjugate of a complex number \[a + bi\] is \[a - bi\]. Complex conjugates play a crucial role in mathematics, especially when dealing with quadratic equations with real coefficients. When a complex root \[a + bi\] is given, and the equation has real coefficients, the other root must be its complex conjugate \[a - bi\]. This property maintains the real nature of the polynomial's coefficients.
Complex conjugates have some useful properties:

• When added together, the imaginary parts cancel out: \[(a+bi) + (a-bi) = 2a\].
• When multiplied, the result is a real number: \[(a+bi)(a-bi) = a^2 + b^2\].

In our exercise, the given solution is \[4 - i\]. Therefore, the other solution must be its complex conjugate, \[4 + i\]. By verifying that \[4 - i\] and \[4 + i\] are complex conjugates, we confirm their correctness as solutions to the quadratic equation with real coefficients.

real coefficients

Real coefficients in a quadratic equation are the constant terms *a*, *b*, and *c* which are real numbers. These coefficients influence the nature of the roots of the equation. When a quadratic equation has real coefficients, it ensures that if one root is complex, the other must be its complex conjugate. This balancing act keeps the coefficients in the form \[ax^2 + bx + c = 0\] strictly real.
Real coefficients have important applications in various fields and help in keeping the solutions of polynomials grounded in real numbers when possible. This is especially vital for physical problems and engineering scenarios where imaginary numbers might not have practical interpretations. For instance, when given a solution as \[4 - i\], recognizing the need for its complex conjugate \[4 + i\] helps in validating the polynomial's real coefficients and ensuring proper formulation of the quadratic equation.