Question

\(2+3 i\) is a solution of a quadratic equation with real coefficients. Find the other solution.

Short answer

The other solution is \(2 - 3i\).

Step-by-step solution

Understand Complex Conjugates

If a quadratic equation with real coefficients has a complex solution, its complex conjugate will also be a solution. The complex conjugate of a number is obtained by changing the sign of the imaginary part. Given the solution is \(2 + 3i\), its complex conjugate is \(2 - 3i\).

Identify the Other Solution

The other solution to the quadratic equation is the complex conjugate of the given solution. Therefore, the other solution is \(2 - 3i\).

Key concepts

quadratic equations

A quadratic equation is a polynomial equation of the second degree. It generally takes the form:

\[ ax^2 + bx + c = 0 \]

where a, b, and c are real coefficients, and x represents the variable. One of the main characteristics of quadratic equations is that their solutions can be real or complex numbers.

The solutions of a quadratic equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, the expression under the square root \(b^2 - 4ac\) is called the discriminant. The nature of the solutions depends on the value of the discriminant:

• If the discriminant is positive, there are two distinct real solutions.


• If it is zero, there is exactly one real solution (a repeated root).


• If it is negative, both solutions are complex conjugates.

In the case of complex solutions, understanding the concept of complex conjugates becomes essential.

complex conjugate

A complex conjugate of a given complex number is formed by changing the sign of its imaginary part. For example, if we take the complex number \(2 + 3i\), its complex conjugate will be \(2 - 3i\).

Complex conjugates hold significant importance when the coefficients of the polynomial are real. This is because, according to the properties of polynomials, if a polynomial with real coefficients has a complex solution, the conjugate of that solution is also a solution.

This property ensures that quadratic equations with real coefficients, having one complex solution, will always have another solution which is the complex conjugate of the first. This principle was utilized to find the solution in our original exercise:

• The given solution is \(2 + 3i\).


• Its complex conjugate is \(2 - 3i\).


• Hence, \(2 - 3i\) is also a solution to the quadratic equation.

Understanding the complex conjugate helps in dealing with quadratic equations that have non-real solutions.

real coefficients

A quadratic equation with real coefficients is one where the coefficients a, b, and c are real numbers. This restriction has interesting implications regarding the nature of the solutions.

When solving such an equation, the potential for complex solutions arises if the discriminant (\(b^2 - 4ac\)) is negative. Real coefficients mandate that complex solutions appear in conjugate pairs.

For example, consider a quadratic equation where the discriminant is negative:

\[ x^2 - 4x + 13 = 0 \]

Here, the discriminant is \((-4)^2 - 4(1)(13) = 16 - 52 = -36\), which is negative. Hence, the solutions are complex. Using the quadratic formula, we get:

\[ x = \frac{4 \pm \sqrt{-36}}{2(1)} = \frac{4 \pm 6i}{2} = 2 \pm 3i \]

Therefore, the solutions are \(2 + 3i\) and \(2 - 3i\), which are complex conjugates. This shows how real coefficients in quadratic equations enforce the occurrence of complex conjugate solutions when the discriminant is negative.