Question

If \(\frac{(4 x+3)^2}{2}=72\) and \(x>0\), then \(x=\)

Short answer

So, when \(x > 0\), the value of \(x = \frac{9}{4}\).

Step-by-step solution

Multiply by 2

To eliminate the fraction, multiply both sides of the equation by 2: \[(4x+3)^2=144\]

Take the square root

Now, take the square root of both sides of the equation to get rid of the square: \[4x+3=\pm\sqrt{144}\] Since \(x>0\), we have: \[4x+3=12\]

Isolate x

Subtract 3 from both sides of the equation to isolate \(x\): \[4x=9\]

Solve for x

Finally, divide both sides of the equation by 4 to solve for \(x\): \[x=\frac{9}{4}\]

Key concepts

Algebra

Algebra is like a puzzle. It deals with symbols and the rules for manipulating these symbols. In this problem, we are using the symbol \(x\) to represent an unknown value. Our task is to find this value. We do so by setting the equation and solving it step by step.

• First, notice the equation starts off with a fraction. To simplify, we multiply both sides by 2, which makes handling the equation easier.
• This is a basic algebraic principle: whatever you do to one side of an equation, you do it to the other to maintain balance.
• Once the fraction is removed, the equation becomes simpler: \((4x+3)^2 = 144\).

Engaging with algebra can feel challenging at first, but it becomes fun as you practice. Every step takes you closer to uncovering the mystery value of \(x\).

Square Root Method

The square root method is an essential tool for solving quadratic equations, especially when the equation involves squares, like \((4x+3)^2\). When you see something squared, you often need to take the square root to simplify. Here's how:

• We start with an equation where something, in this case \(4x+3\), is squared.
• To remove this squaring, we take the square root of both sides of the equation. This step transforms \((4x+3)^2 = 144\) into \(4x+3 = \pm 12\).
• The \(\pm\) symbol reflects that both positive and negative roots are possible because squaring either gives the same result.

However, since \(x > 0\) in this problem, we only consider the positive root \(4x+3=12\). This method is powerful because it simplifies equations, making it easier to find solutions.

Isolation of Variables

Isolation of variables is the next crucial step in solving for \(x\). It means getting \(x\) by itself on one side of the equation. This process is like peeling away layers of an onion to reveal what's inside. Here's how it's done:

• In the equation \(4x+3=12\), our goal is to get \(x\) alone. We start by subtracting 3 from both sides, which leaves us with \(4x=9\).
• Next, we divide by 4, the coefficient of \(x\), to isolate \(x\). This division results in \(x = \frac{9}{4}\).
• Always remember, isolating \(x\) involves undoing arithmetic operations through their reverse actions: subtraction for addition, division for multiplication, and so on.

By isolating the variable, we can solve the equation and find the value of \(x\). This process highlights the orderliness in algebraic structures.