Problem 76 - Elementary Algebra

Question

Problem: Solve: \(x^{2}=324\) Incorrect Answer: \(\begin{aligned} x^{2} &=324 \\ x &=\sqrt{324} \\ x &=18 \end{aligned}\)

Step-by-Step Solution

Answer

x = 18 or x = -18

Step 1: Determine the Equation

Identify the given equation, which is \(x^2 = 324\).

Step 2: Apply the Square Root

Take the square root of both sides of the equation. This leads to \(\begin{aligned} \sqrt{x^2} &= \sqrt{324} \end{aligned})\).

Step 3: Solve for x

Simplify the equation on both sides to find \(x\): \(\begin{aligned} x &= \pm 18 \end{aligned})\).

Key Concepts

Square Roots

To understand how to solve the quadratic equation \(x^2 = 324\), we first need to discuss square roots. The square root of a number is a value that, when multiplied by itself, gives the original number.

For example, the square root of 324 is because \(18 \times 18 = 324\). However, the square root is not just positive. A number can also have a negative square root. This is because multiplying two negative numbers also results in a positive product. So, \(-18 \times -18 = 324\) too.

Thus, any time we take a square root of a number, we must consider both the positive and the negative values.

Positive and Negative Solutions

When solving the equation \(x^2 = 324\), it is essential to remember that there are two solutions. Not just the positive root but also the negative one.

When we take the square root of both sides, we have:

\sqrt{x^2} = \sqrt{324}

This becomes: \pm x = 18

The symbol \pm denotes that x can be both +18 and -18. Thus, we have:

x = 18 or x = -18

Remember always to include both solutions when dealing with squares since real numbers have both positive and negative square roots.

Simplifying Equations

To solve the given problem efficiently, it is necessary to simplify the quadratic equation correctly. Let’s go through it step by step:

• Identify the equation: \(x^2 = 324\).
• Take the square root on both sides: \sqrt{x^2} = \sqrt{324}.
• Simplify the equation: x = \pm 18.

Simplification is essential because it ensures we get the correct solution. In this case, simplifying tells us that x could be either positive or negative18, providing two possible values for x.