Problem 13 - Algebra for College Students

Question

For each given pair of numbers find a quadratic equation with integral coefficients that has the numbers as its solutions. See Example 1. $$i \sqrt{2},-i \sqrt{2}$$

Step-by-Step Solution

Answer

The quadratic equation is \[ x^2 + 2 = 0 \].

Step 1 - Understand the given numbers

The given numbers are the roots of the quadratic equation: \(i \sqrt{2} \) and \(-i \sqrt{2}\).

Step 2 - Form the polynomial equation

Use the fact that if \( x_1 \) and \( x_2 \) are roots, then the polynomial can be written as \( (x - x_1)(x - x_2) \). So, the equation becomes \[ (x - i \sqrt{2})(x + i \sqrt{2}). \]

Step 3 - Simplify using the difference of squares

Recognize that \((a - b)(a + b) = a^2 - b^2\). Therefore, \[ (x - i \sqrt{2})(x + i \sqrt{2}) = x^2 - (i \sqrt{2})^2 \]

Step 4 - Substitute and simplify further

Replace \((i \sqrt{2})^2 \) with \(-2 \): \[ x^2 - (-2) = x^2 + 2 \] Thus, the quadratic equation is \[ x^2 + 2 = 0 \]

Key Concepts

Integral Coefficients

In mathematics, integral coefficients refer to the coefficients in an equation that are whole numbers. For quadratic equations, ensuring the coefficients are integers simplifies the process and makes it easier to solve and understand.
When forming a polynomial from given roots, we often need to convert any complex terms into a form that yields integral coefficients.
For the given problem, our roots were complex: \(i \sqrt{2} \) and \(-i \sqrt{2}\). Through the steps, we saw that transforming these into the polynomial expression resulted in the quadratic equation with integral coefficients: \[ x^2 + 2 = 0 \].
This transformation is essential because it avoids any irrational or complex values in the coefficients, providing a simpler and more manageable equation.

Complex Roots

Complex roots appear when the solutions to an equation involve the imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\). For quadratic equations, the roots can be real or complex.
Given the roots from the problem, \(i \sqrt{2}\) and \(-i \sqrt{2}\), these represent the imaginary parts common in complex roots.
Using these complex roots, we formed the polynomial:

\t
• Start with the factors \( (x - i \sqrt{2})(x + i \sqrt{2}) \).
\t
• Next, we apply the difference of squares identity to simplify it.

The concept of complex roots is crucial in understanding how solutions to quadratic equations can extend beyond the real number line, adding depth to the study of algebra.

Difference of Squares

The 'difference of squares' is a powerful algebraic tool used to simplify expressions where two squares are subtracted. It follows the identity: \[ (a - b)(a + b) = a^2 - b^2\]. This identity helps in breaking down complex expressions into simpler forms.
In the provided problem, recognizing the difference of squares allowed us to efficiently simplify:

\t
• \[ (x - i \sqrt{2})(x + i \sqrt{2}) = x^2 - (i \sqrt{2})^2\]
\t
• This translates to \[ x^2 - (-2) = x^2 + 2\]

By leveraging the difference of squares, we turned a potentially complicated equation into a manageable quadratic equation with integral coefficients.
This method is widely used across algebra to simplify and solve equations with ease.