Problem 96 - Algebra for College Students

Question

Cooperative learning. Work with a group to write a quadratic equation that has each given pair of solutions. a) \(3+\sqrt{5}, 3-\sqrt{5}\) b) \(4-2 i, 4+2 i\) c) \(\frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\)

Step-by-Step Solution

Answer

a) \ x^2 - 6x + 4 = 0 \; b) \ x^2 - 8x + 20 = 0 \; c) \ x^2 - x + 1 = 0 \.

Step 1: Understand the Problem

Given pairs of solutions, write quadratic equations that have these solutions. Remember, if \(x_1\) and \(x_2\) are the solutions, the quadratic equation can be written as \( (x - x_1)(x - x_2) = 0 \).

Step 2: Write the Quadratic Equation for Part (a)

Given solutions are \(3+\sqrt{5}\) and \(3-\sqrt{5}\).The quadratic equation with these roots can be written as \( (x - (3+\sqrt{5}))(x - (3-\sqrt{5})) = 0 \).Simplify the product using the difference of squares formula:\[ (x - (3+\sqrt{5}))(x - (3-\root{5})) = (x-3)^2 - (\sqrt{5})^2 = x^2 - 6x + 4 = 0 \]

Step 3: Write the Quadratic Equation for Part (b)

Given solutions are \(4-2i\) and \(4+2i\).The quadratic equation with these roots can be written as \( (x - (4-2i))(x - (4+2i)) = 0 \).Simplify the product using the difference of squares formula:\[ (x - 4 + 2i)(x - 4 - 2i) = (x-4)^2 - (2i)^2 = x^2 - 8x + 20 = 0 \]

Step 4: Write the Quadratic Equation for Part (c)

Given solutions are \( \frac{1+i\sqrt{3}}{2} \) and \( \frac{1-i\sqrt{3}}{2} \).The quadratic equation with these roots can be written as \( \left(x - \frac{1+i\sqrt{3}}{2}\right)\left(x - \frac{1-i\root{3}}{2}\right) = 0 \).Simplify the product using the difference of squares formula:\[ \left(2x - (1+i\sqrt{3})\right)(2x - (1-i\sqrt{3})) = (2x-1)^2 - (i\sqrt{3})^2 = 4x^2 - 2x + 1 = x^2 - x + 1 = 0 \]

Key Concepts

complex roots

In quadratic equations, complex roots occur when the solutions include imaginary numbers. Imaginary numbers are multiples of the imaginary unit 'i', where \(i = \sqrt{-1}\). These roots often appear in pairs of the form \(a + bi\) and \(a - bi\), where 'b' is a real number and 'i' is the imaginary unit.
When you have complex roots, the quadratic equation can still be written by noticing that the product of the solutions gives the equation. To form an equation from the solutions:

• Given the solutions \(x_1 = 4 - 2i\) and \(x_2 = 4 + 2i\), you can write the equation as: \((x - x_1)(x - x_2) = 0\).
• Simplify using the difference of squares formula: \((x - 4 + 2i)(x - 4 - 2i) = (x - 4)^2 - (2i)^2\).
• This becomes: \(x^2 - 8x + 20 = 0\).

Complex roots add a beautiful symmetry to quadratic equations and reveal the algebraic richness in mathematics.

difference of squares

The difference of squares is a helpful method for simplifying quadratic equations. It is based on the identity: \((a - b)(a + b) = a^2 - b^2\). This identity is particularly useful when simplifying products of expressions such as \((x - (a + \sqrt{b}))(x - (a - \sqrt{b}))\).
For instance, given solutions like \(3 + \sqrt{5}\) and \(3 - \sqrt{5}\), you can use the difference of squares to simplify the product:

• Start by writing the product: \((x - (3 + \sqrt{5}))(x - (3 - \sqrt{5}))\).
• Simplify using the difference of squares identity: \((x - 3)^2 - (\sqrt{5})^2 = x^2 - 6x + 4\).

This powerful algebraic technique allows you to simplify and solve quadratic equations more easily. Once you identify a pair of conjugate roots, you can quickly use the difference of squares to find the corresponding quadratic equation.

simplifying quadratic equations

Simplifying quadratic equations involves writing the equation in a standard form, usually \(ax^2 + bx + c = 0\). By breaking down more complicated expressions and combining like terms, you can simplify quadratics to make them easier to solve.
Consider the solution pair \(\frac{1 + i \sqrt{3}}{2}\) and \(\frac{1 - i \sqrt{3}}{2}\) for a quadratic equation. To simplify this, you would:

• Write it in product form: \((x - \frac{1 + i \sqrt{3}}{2})(x - \frac{1 - i \sqrt{3}}{2}) = 0\).
• Combine the terms and use the difference of squares formula: \(\left(2x - (1 + i \sqrt{3})\right)(2x - (1 - i \sqrt{3})) = (2x - 1)^2 - (i \sqrt{3})^2\).
• Further simplify to \(4x^2 - 2x + 1 = 0\).

Using these steps, you transfer the complex roots to a simplified quadratic form that is easier to handle algebraically. Simplifying these equations can uncover different properties and applications within mathematics.