Question

Show there is no quadratic equation \(a x^2+b x+c=0\) such that \(a, b\), and \(c\) are real numbers and \(3 i\) and \(-2 i\) are solutions.

Short answer

The roots \(3i\) and \(-2i\) are not conjugates, hence there is no quadratic equation \(a x^2+b x+c=0\) (where \(a, b, c\) are real), with roots \(3 i\) and \(-2 i\). Consequently, there exists no such quadratic equation.

Step-by-step solution

Identify the Roots

The task gives two roots of the quadratic equation, namely \(3 i\) and \(-2 i\). Note that these roots are purely imaginary.

Check for Conjugate Roots

Now we need to understand if these roots could pair to form a valid pair of conjugate roots. Conjugate roots are always in the form \(a + bi\) and \(a - bi\). If \(3 i = a + bi\), then the conjugate would be \(a - bi\). Same for \(-2 i\). However, there is no real part (represented by 'a') in these given roots. Thus, neither of the two roots \(-2 i\) and \(3 i\) have a conjugate.

Conclusion

Since the given roots are not conjugate pairs and the coefficients of the quadratic equation are real, it's not possible for both \(3 i\) and \(-2 i\) to be roots of the same quadratic equation with real coefficients. Therefore, there exists no quadratic equation \(a x^2+b x+c=0\) with real coefficients \(a, b, c\) such that \(3 i\) and \(-2 i\) are solutions.

Key concepts

Imaginary Numbers

Imaginary numbers are numbers that are used to measure quantities with no real counterpart. Often denoted by the symbol 'i,' an imaginary number is defined as the square root of -1. Imaginary numbers are crucial in solving equations that cannot be satisfied with real numbers alone.
For example, the number 3i represents an imaginary number, where 'i' is the imaginary component. Imaginary numbers often appear in pairs called complex numbers, which include both real and imaginary parts. In this context, the solution of a quadratic equation might include imaginary numbers, especially when the coefficients are real but the discriminant (from the quadratic formula) is negative, indicating no real roots exist.

Conjugate Roots

Conjugate roots play an essential role in polynomial equations, especially those with real coefficients. If a polynomial with real coefficients has non-real complex or imaginary roots, they must appear in conjugate pairs.
Conjugate pairs have the form \(a + bi\) and \(a - bi\), where 'a' is the real part and 'b' is the imaginary part. The key idea here is that the imaginary part is identical, but the sign between the real and imaginary parts differs.
For example, if you have a root \(3i\), its conjugate would be \(-3i\) because the real component 'a' is zero. The exercise asks whether \(3i\) and \(-2i\) can be roots of a quadratic equation with real coefficients, but they are not valid conjugates because neither shares a common real part 'a.' This lack of a common real part means they cannot be conjugate roots, ruling them out as possible paired roots for the given polynomial equation.

Quadratic Formula

The quadratic formula is a crucial tool in finding the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula allows us to find solutions by substituting the values of \(a\), \(b\), and \(c\) and computing the discriminant \(b^2 - 4ac\).
- If the discriminant is positive, the equation has two real roots.- If it's zero, there is one real double root.- If negative, the roots are complex or imaginary.
In the problem, the imaginary roots \(3i\) and \(-2i\) indicate a scenario where the discriminant is negative. However, real coefficients require conjugate pairs like \(3i\) and \(-3i\) for the standard quadratic formula to apply. Thus, \(3i\) and \(-2i\) cannot be roots of the same quadratic with real coefficients, as no real set of coefficients meets this condition within the quadratic formula framework.

Complex Numbers

Complex numbers integrate both real and imaginary numbers within a single number system. They are generally depicted in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit representing \(\sqrt{-1}\).
Complex numbers can be visualized on a complex plane, which enhances understanding and solutions for many engineering and physics problems.
- The number \(3i\) is a complex number with '0' as the real part and '3' as the imaginary part.- Likewise, \(-2i\) has no real part and \(-2\) as the imaginary part.
In solving quadratic equations, identifying roots within this larger number system can offer a solution even when real roots don’t exist. However, due to the requirements of real coefficients yielding conjugate roots, \(3i\) and \(-2i\) do not satisfy this pairing requirement.The absence of a matching real part 'a' affects them being valid solutions for the same polynomial equation. Complex numbers explain why a quadratic like the one described in the exercise lacks a real solution with the given roots.