Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two quadratic equations having the given solutions. (There are many correct answers.) $$3+4 i, 3-4 i$$

Step-by-Step Solution

Answer

Two quadratic equations having the given solutions are \(x^2 - 6x + 25 = 0\) and \(k*(x^2 - 6x + 25) = 0\), where \(k\) is any non-zero real number.

Step 1: Understanding Complex Conjugate Roots

In any quadratic equation if the coefficients are real numbers and roots are complex, then they exist as a pair of Complex conjugate. Like in our problem, we are given roots as \(3+4i\) and \(3-4i)\). If a+bi is one root then a-bi is the other root. So, our roots are a pair of complex conjugate

Step 2: Apply Viète's Formulas

Viète's formulas allows creating a quadratic equation from its roots. To apply this method, the roots \(a = 3+4i\) and \(b = 3-4i\) are inserted into the formula \(x^2 - (a+b) x + ab = 0\).

Step 3: Calculate the Sum and Product of the Roots

The sum of the roots \(a + b = (3+4i) + (3-4i) = 6\) and the product \(ab = (3+4i) * (3 - 4i) = 9+16 = 25\)

Step 4: Construct the Equation

Substitute the sum and product of roots into the formula, we get \(x^2 - 6x + 25 = 0\)

Key Concepts

Complex Conjugate Roots

Complex conjugate roots are pairs of complex numbers associated with quadratic equations that have real number coefficients. When you have a complex root in the form of \(a + bi\), its complex conjugate will be \(a - bi\). They are a type of symmetry seen in complex number mathematics.
Understanding these roots is essential in algebra, particularly when dealing with equations that do not have straightforward real number solutions.
For instance, in the problem given, the roots \(3 + 4i\) and \(3 - 4i\) are complex conjugates. Notice how the imaginary parts are equal in magnitude and opposite in sign. This unique property makes complex conjugates crucial for forming quadratic equations with specific solutions.
Another important aspect of complex conjugate roots is their role in ensuring that when they appear, the resulting polynomial remains a real-valued function. This is vital when solving quadratic equations with real coefficients.

Viète's Formulas

Viète's formulas are powerful mathematical tools that link the coefficients of a polynomial to sums and products of its roots. Particularly for quadratic equations, these formulas are essential in deriving the equation from given roots without direct multiplication.
Viète's formulas can be expressed for a quadratic equation \(x^2 - (a+b)x + ab = 0\), where \(a\) and \(b\) are roots. Here, \(a+b\) gives the negative coefficient of \(x\), and \(ab\) results in the constant term.
In our scenario, the roots are \(3 + 4i\) and \(3 - 4i\). According to Viète's, their sum is \((3 + 4i) + (3 - 4i) = 6\), and the product is \((3 + 4i)(3 - 4i) = 9 + 16 = 25\).

• Sum of roots \(a+b\) leads to the\(-6x\) term in the expression.
• Product \(ab\) gives the constant term \(25\).

By employing Viète’s formulas, we formulate the quadratic equation \(x^2 - 6x + 25 = 0\) easily.

Complex Numbers

Complex numbers extend our familiar number system by introducing the imaginary unit \(i\), where \(i^2 = -1\). These numbers take the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
They are crucial in mathematics because they expand the solutions of equations beyond the real number limitations. For instance, the equation \(x^2 + 1 = 0\) has no real solutions, but with complex numbers, solutions \(x = i\) and \(x = -i\) are possible.
Aside from providing complete solutions for polynomial equations, particularly those involving square roots of negative numbers, complex numbers also play vital roles in other fields. They are extensively used in engineering, physics, and computer science for handling sinusoidal and oscillatory problems.
Understanding complex numbers involves both their algebraic properties and their geometric representation. In the complex plane, every complex number corresponds to a point, aiding in visual comprehension and further manipulation of mathematical problems.