Problem 31 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=64 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 8 \) and \( x = -8 \).

Step 1: Square Root Both Sides

We take the square root of both sides of the equation: \( \sqrt{x^{2}} = \sqrt{64} \). This is because the opposite operation of squaring is taking the square root.

Step 2: Solve for x

This yields two possible equations: \( x = 8 \) and \( x = -8 \). This is because both 8 and -8 squared will give us 64.

Step 3: Check Solutions

To ensure these are the correct solutions, we can check by substituting them back into the original equation: \( (8)^{2} = 64 \) and \( (-8)^{2} = 64 \). Both of these checks confirm that 8 and -8 are indeed the correct solutions.

Key Concepts

Square Root Method

The square root method is a simple and effective way to solve quadratic equations, specifically when the equation is in the form of \( x^2 = c \). This type of equation is typically solved by isolating the variable on one side and taking the square root of both sides.

• When dealing with \( x^2 = c \), the goal is to isolate \( x \) by taking the positive and negative square roots of \( c \) because squaring either positive or negative numbers give the same result.
• It is important to remember that the square root has two values: a positive and a negative root. Thus, the required step is writing both \( x = \sqrt{c} \) and \( x = -\sqrt{c} \).
• For example, with the equation \( x^2 = 64 \), taking the square root of both sides leads to two potential solutions: \( x = 8 \) and \( x = -8 \).

This method is especially useful for equations where squaring is isolated and there is no linear component such as a term with just \( x \). It provides a direct route to find solutions without needing to use more complex methods like factoring or using the quadratic formula.

Solving Equations

When solving equations, particularly quadratic ones, finding the value of the variable involves steps that simplify the equation to make solving straightforward. Quadratic equations like \( x^2 = 64 \) come with their own set of strategies.

• First, ensure the equation is simplified so that all terms involving \( x \) are on one side and constants on the other.
• For \( x^2 = 64 \), taking the square root method makes it simple since there's no additional \( x \) term to complicate the process.
• The solution is confirmed by substituting the found values back into the original equation. This confirms that squaring the solutions yields the original value on the other side of the equation.

This approach is efficient because it limits the steps needed to solve and verify solutions through direct substitution—simplifying both solving and checking processes.

Real Solutions

Real solutions refer to solutions that are real numbers, as opposed to complex or imaginary numbers. In the context of quadratic equations, these solutions exist on the real number line.

• For the equation \( x^2 = 64 \), the solutions \( x = 8 \) and \( x = -8 \) are both real numbers because squaring them yields a positive number, which remains in the realm of real numbers.
• Real solutions are crucial because they represent values that can exist in practical, observable contexts, unlike imaginary numbers which cannot be visualized on a standard number line.
• Quadratic equations like \( x^2 = 64 \) that yield real solutions are particularly simple as they do not require dealing with square roots of negative numbers which introduce imaginary units \(i\).

Understanding that equations can produce real solutions helps in predicting the nature of their graphs and in comprehending the problem's practical applications, if any.