Problem 71 - Algebra 2

Question

Write a quadratic equation with the given solutions. \(\frac{5+i \sqrt{3}}{2}, \frac{5-i \sqrt{3}}{2}\)

Step-by-Step Solution

Answer

The quadratic equation with these roots is \(x^2 - 5x + 4 = 0\).

Step 1: Identify the roots

The roots provided to us are \(\frac{5+i \sqrt{3}}{2}\) and \(\frac{5-i \sqrt{3}}{2}\)

Step 2: Formulate the quadratic equation

Using these roots in the formulation \(a(x - p)(x - q) = 0\) gives us \(a[(x - \frac{5+i \sqrt{3}}{2})(x - \frac{5-i \sqrt{3}}{2})] = 0\)

Step 3: Expand the equation

The equation expands to \(a[(x^2 - x\frac{5+i \sqrt{3}}{2} - x\frac{5-i \sqrt{3}}{2} + (\frac{5+i \sqrt{3}}{2}) (\frac{5-i \sqrt{3}}{2})] = 0\) which simplifies to \(a[x^2 - x(5) + 4] = 0\)

Step 4: Write the final equation

Let's consider \(a = 1\) (as coefficient of \(x^2\) cannot be zero) for simplicity, the final equation becomes \(x^2 - 5x + 4 = 0\).

Key Concepts

Complex Roots

In mathematics, complex roots arise when attempting to solve a quadratic equation that does not cross the x-axis on a graph. These roots include an imaginary number, represented as 'i', where \(i^2 = -1\). For example, the numbers \(\frac{5+i \sqrt{3}}{2}\) and \(\frac{5-i \sqrt{3}}{2}\) are complex roots. Complex roots come in conjugate pairs when coefficients of the polynomial are real numbers. This ensures that the imaginary parts are opposite in sign, providing anchor points for the polynomial's symmetry on the complex plane. Understanding complex roots is useful for visualizing and solving polynomials in real-world scenarios like electrical engineering or quantum physics.

Solving Quadratic Equations

Solving quadratic equations is a fundamental algebraic skill. A quadratic equation is typically in the form \(ax^2+bx+c=0\). The solutions to such equations are the values of \(x\) that satisfy this equation. To solve these, you can use different methods:

• Factoring: Express the quadratic in a product of two binomials.
• Using the Quadratic Formula: Apply the formula \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{2a}\).
• Completing the Square: Rearrange the equation into a perfect square binomial.

For the equation \(x^2 - 5x + 4 = 0\), these roots can be found using any of these methods. However, when given the roots directly, as in this exercise, you can simply plug them into the formulation to find the quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, regardless of whether they are real or complex. This formula is derived from the process of completing the square and is uniquely advantageous because it works universally and doesn't require the equation to be factored easily or have straightforward integer solutions. For a quadratic equation \(ax^2 + bx + c = 0\), the quadratic formula is given by:\[ x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{2a}\]Here, \(b^2 - 4ac\) is called the discriminant. It determines the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two real and distinct roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex and form a conjugate pair.

In the current scenario, using the quadratic formula would show that the discriminant is negative, confirming the presence of complex roots. This illustrates the flexibility and importance of the quadratic formula in handling complex numbers in algebra.