Problem 70 - Beginning and Intermediate Algebra

Question

Solve for \(x .\) Assume that a and \(b\) represent positive real numbers. \(x^{2}=4 b\)

Step-by-Step Solution

Answer

x = \pm 2\sqrt{b}

Step 1 - Understand the Problem

We are given the equation \(x^{2} = 4b\) and need to find the value of \(x\). Since \(a\) and \(b\) are positive real numbers, \(b > 0\).

Step 2 - Isolate the Variable

To isolate \(x\), we need to eliminate the exponent. To do this, take the square root of both sides of the equation. We obtain \(\sqrt{x^2}\) on the left and \(\sqrt{4b}\) on the right.

Step 3 - Solve for \x\

Taking the square root of \(x^{2}\) gives us \(x\) on the left, and the square root of \(4b\) is \(\sqrt{4b} = 2\sqrt{b}\). Therefore, \(x = \pm 2\sqrt{b}\).

Key Concepts

isolating variables

Isolating variables means getting the main variable you are solving for alone on one side of the equation. This is often the first step when working with equations. In our example, we need to isolate the variable \( x \) in the equation \( x^{2}=4b \).
To isolate \( x \), we can apply operations that undo what has been done to \( x \). Here, \( x \) is squared, so to undo the squaring, we use the opposite operation: taking the square root. This helps us to isolate the variable \( x \), making it easier to solve the equation.

square roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \).
In our quadratic equation \( x^{2}=4b \), we take the square root of both sides to isolate \( x \).
Taking the square root of \( x^{2} \) results in \( \text{\textpm} x \). This is because both positive and negative values squared will give us a positive result. For the right side, we calculate \( \text{\textpm} \text{\textbackslash} sqrt{4b} \).
Simplifying further, \( 4 \) is a perfect square, and its square root is 2. Thus, we get \( \text{\textpm} 2\text{\textbackslash} sqrt{b} \). So, \( x = \text{\textpm} 2\text{\textbackslash} sqrt{b} \text{ represents both positive and negative solutions for x. } \)

positive real numbers

Positive real numbers are numbers that are greater than zero and are not imaginary. They include all the decimal and fractional values.
In our context, it is given that \( b \) is a positive real number, meaning \( b > 0 \). This is important because it guarantees that \( \text{\textbackslash} sqrt{b} \) is a real number and not an imaginary number.
Since \( b \) is assumed to be positive, we are ensured valid real number solutions for \( x \). This helps avoid complications, keeping the problem straightforward and solvable with basic algebraic techniques.