Problem 38 - Elementary Algebra

Question

For the following problems, solve for the indicated variable. $$ x^{2}=4 a^{2}, \text { for } x $$

Step-by-Step Solution

Answer

Question: Solve the equation $$x^2 = 4a^2$$ for x. Answer: x = 2a, -2a.

Step 1: Isolate x

In the given equation, $$x^2 = 4a^2$$, x is already isolated on the left side. However, to solve for x, we must take the square root of both sides.

Step 2: Take the square root of both sides

To solve for x, apply square root on both sides of the equation: $$ \sqrt{x^2} = \sqrt{4a^2} $$

Step 3: Simplify the equation

Now we will simplify the equation and express the result: $$ x = \pm 2a $$ So, the solution for x in terms of a is: $$ x = 2a, -2a $$

Key Concepts

Isolating Variables

One fundamental step in solving equations is isolating the variable of interest. In our problem, the equation is presented as \(x^2 = 4a^2\), which aims to solve for \(x\). Interestingly, in this instance, \(x\) is already isolated on one side of the equation. This means that it's already set apart from other terms, making it easier to work on finding its value. Isolating variables involves moving all other components of the equation away from the variable you are solving for. You may need to perform operations like addition, subtraction, multiplication, or division to move terms around. For example:- If the equation were \(2x + 6 = 10\), you would first subtract 6 from both sides to get \(2x = 4\), and then divide by 2 to isolate \(x\), leading to \(x = 2\).In the given exercise, since the variable is already isolated, this reduces the workload, leaving merely the computation steps to find the solution.

Taking Square Roots

Taking the square root is a pivotal step when dealing with quadratic equations, like \(x^2 = 4a^2\). The square root is essentially the reverse operation of squaring a number.To solve for \(x\), you need to eliminate the square by applying the square root to both sides of the equation. Here's how it's done:- When you take the square root of \(x^2\), you are left with \(x\). This is because the square root function and the square cancel each other out.- Similarly, when you take the square root of \(4a^2\), it simplifies to \(2a\) because \(\sqrt{4} = 2\) and \(\sqrt{a^2} = a\). Remember, taking a square root will yield both positive and negative solutions. This is because either squared will give the original positive value.So, after applying the square root, you will have:\[x = \pm 2a\]The \(\pm\) symbol indicates that \(x\) could be either \(2a\) or \(-2a\). This dual nature occurs because both \((2a)^2\) and \((-2a)^2\) return \(4a^2\).

Equations Involving Squares

Equations with squares are common in algebra, especially quadratic equations like \(x^2 = 4a^2\). These types of equations typically involve a variable raised to the power of two, and solving them often requires special techniques.When solving square equations, recognition of their format is key. Such equations may appear in vast scenarios, such as calculating areas or modeling physical phenomena. Here's what to focus on:- **Structure**: Understanding that these equations are in a form where the variable is squared (such as \(x^2\) or \(a^2\)) is crucial.- **Solving Method**: As discussed earlier, using square roots is a standard method to "unsquare" the variable. Always remember the solution requires checking for both positive and negative roots.Approaching such equations consolidates not just algebraic knowledge but also enhances problem-solving skills as students recognize patterns and apply methods efficiently. By mastering equations involving squares, students can handle more complex tasks that mathematics presents.