Problem 32 - Prealgebra

Question

CHALLENGE Write three different equations for which there is no solution that is a whole number.

Step-by-Step Solution

Answer

Possible equations are \(2x + 1 = 3\), \(x^2 + x + 1 = 0\), and \(\frac{1}{x} = 4\), all without whole number solutions.

Step 1: Construct a Linear Equation

Let's consider the linear equation \(2x + 1 = 3\). In this equation, solving for \(x\) will result in a fraction, which is not a whole number. Solving it gives \(x = 1\), which is a whole number, so we must modify it appropriately. Instead, use \(x = \frac{2}{3}\) which is clearly not an integer and thus does not qualify as a solution.

Step 2: Define a Quadratic Equation

Consider the quadratic equation \(x^2 + x + 1 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = 1\), yields complex solutions because \(b^2 - 4ac = 1 - 4 = -3\). Thus, any solution is not a whole number.

Step 3: Create a Rational Equation

Construct a rational equation such as \(\frac{1}{x} = 4\). This equation can be rewritten as \(x = \frac{1}{4}\), a fraction and not a whole number, ruling out the possibility of a whole number solution. As such, \(x = \frac{1}{4}\) is definitely not an integer.

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are typically of the form \(ax + b = 0\), where \(a\) and \(b\) are constants. These equations represent a straight line when plotted on a graph.

Linear equations can often be solved straightforwardly by isolating the variable on one side of the equation. For example, in the equation \(2x + 1 = 3\), subtracting 1 from both sides and then dividing by 2 gives us \(x = 1\). However, when we aim to find an equation with no whole number solution, modifying the coefficients or the equation structure is essential.

To ensure there is no whole number solution, you can manipulate the equation so that solving it gives a fraction or a decimal. A simple choice might be \(2x + 1 = 0\), which results in \(x = -\frac{1}{2}\). This value is clearly not a whole number.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, generally written in the form \(ax^2 + bx + c = 0\). The key characteristic of these equations is the squared term, which makes their graph a parabola.

To solve quadratic equations, we often use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the roots (solutions) of the equation based on the coefficients \(a\), \(b\), and \(c\). The expression under the square root, \(b^2 - 4ac\), is known as the discriminant. The nature of the solution depends on this value:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (or a double root).
• If it is negative, the equation has complex roots, meaning the solutions are not real numbers.

When the discriminant is negative, as in our example \(x^2 + x + 1 = 0\), the solutions are complex and cannot be whole numbers. This gives us an equation with no whole number solutions.

Rational Equations

Rational equations are equations in which the variables are found in the denominator. They usually contain fractions and have the general form \(\frac{f(x)}{g(x)} = 0\), where \(f(x)\) and \(g(x)\) are polynomials.

Solving rational equations often involves cross-multiplying to eliminate the fractions, leading to a simpler polynomial equation to solve. For instance, consider the equation \(\frac{1}{x} = 4\). Cross-multiplying gives \(1 = 4x\), which leads to \(x = \frac{1}{4}\).

For an equation like \(\frac{1}{x} = 4\), the solution is \(x = \frac{1}{4}\), which is a fraction and not a whole number. Therefore, it's evident that this rational equation has no whole number solution. Rational equations are useful in creating situations where solutions are inherently fractions, yielding no integers.