Problem 58 - Algebra for College Students

Question

Find all real number solutions for each equation. $$x^{2}-1=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Understand the Equation

We have the equation \(x^2 - 1 = 0\). This is a quadratic equation, which means we need to find the values of \(x\) that satisfy the equation, where the squared term is 1.

Step 2: Rearrange the Equation

Add 1 to both sides of the equation to isolate \(x^2\). This gives us \(x^2 = 1\).

Step 3: Solve for x

To find \(x\), take the square root of both sides of the equation \(x^2 = 1\). Remember that squaring has two possible outputs, one positive and one negative, so \(x = \pm \sqrt{1}\).

Step 4: Simplify the Square Root

The square root of 1 is 1, so when we simplify, we find \(x = \pm 1\).

Step 5: Conclude the Solutions

The real number solutions to the equation \(x^2 - 1 = 0\) are \(x = 1\) and \(x = -1\).

Key Concepts

Real Number Solutions

Real number solutions refer to potential values of \(x\) that satisfy an equation in the set of real numbers. In the context of the quadratic equation \(x^2 - 1 = 0\), we seek all real numbers \(x\) that make the equation true. Here, the real numbers are all the rational and irrational numbers, which can be plotted on a number line. When we say a quadratic equation has real number solutions, it means that we can find at least one point (or number) on the number line that satisfies the equation. For \(x^2 - 1 = 0\), the solutions \(x = 1\) and \(x = -1\) indicate that these are the real numbers that work. If a quadratic equation has no real solutions, it means solutions do not exist in the set of real numbers, often implying complex or imaginary numbers.

Solving Equations

Solving equations is the process of finding values that satisfy mathematical expressions. With quadratic equations like \(x^2 - 1 = 0\), the task is to find the roots or solutions for the unknown variable \(x\). Quadratic equations can typically be in the form \(ax^2 + bx + c = 0\), but they can also be simplified or rearranged. To solve quadratic equations:

• You can rewrite or rearrange the terms to isolate the variable \(x\) or \(x^2\). This helps simplify the calculation.
• Once isolated, use algebraic techniques like factoring, completing the square, or using the quadratic formula to find solutions.

In our example \(x^2 = 1\), we simplified the equation to isolate \(x^2\), and then applied the square root method to solve for \(x\). This is an efficient way when equations are in a simple form.

Square Root Method

The square root method is a quick method for solving equations of the form \(x^2 = a\). It involves 'undoing' the square by taking the square root of both sides of the equation. This method is particularly useful for equations that can be easily rewritten in the form \(x^2 = a\). Steps for using the square root method:

• First, isolate \(x^2\) on one side of the equation.
• Next, apply the square root operation to both sides. Remember that every positive number has two square roots: a positive and a negative.

In the exercise \(x^2 = 1\), taking the square root of both sides gives \(x = \pm 1\). This shows both the positive and negative roots, ensuring all real number solutions are found. This method is straightforward but can only be applied when \(x^2\) is isolated and the right side is a perfect square.