Problem 57 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (4 x-3)^{2}=9 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3}{2} \) and \( x = 0 \).

Step 1: Isolate the Squared Term

The equation is already isolated with the squared term on one side:y (4x - 3)^2 = 9

Step 2: Apply the Square Root Property

Take the square root of both sides of the equation to eliminate the exponent. Remember to include both the positive and negative roots:\[(4x - 3) = \pm \sqrt{9}\]

Step 3: Simplify the Square Root

The square root of 9 simplifies to 3. So we have two possible equations:\[4x - 3 = 3\] and \[4x - 3 = -3\]

Step 4: Solve for x

Solve each equation separately to find the values of x:1. For the equation \[4x - 3 = 3\]: \[4x = 6\] \[x = \frac{6}{4}\] \[x = \frac{3}{2}\]2. For the equation \[4x - 3 = -3\]: \[4x = 0\] \[x = 0\]

Step 5: List the Solutions

The solutions for the equation are:\[x = \frac{3}{2}\] and \[x = 0\]

Key Concepts

Square Root Property

The square root property is a powerful tool for solving quadratic equations, especially when the equation is in the form \((ax - b)^2 = c\). Here's how it works:
When you see an equation like this, you can directly take the square root of both sides. The square root property states that if \(u^2 = v\), then \(u = \pm \sqrt{v}\).
Don't forget to consider both the positive and negative roots. This is important because squaring either a positive or negative number will give you the same result.

Simplifying Radicals

Simplifying radicals is the process of breaking down the square root into its simplest form. In this exercise, we encountered the square root of 9: \(\sqrt{9}\).
The square root of any number can often be simplified if the number under the root is a perfect square. For example:

• The square root of 9 is 3 because 3 times 3 equals 9.
• Similarly, \(\sqrt{16}\ = 4\) because 4 times 4 equals 16.

Knowing how to simplify radicals will make solving equations easier.

Isolating the Squared Term

Before applying the square root property, you need to isolate the squared term. This means you need to have the term with the square on one side of the equation and the constant on the other.
In the given problem, \((4x - 3)^2 = 9\), the term \(4x - 3\) is already squared and isolated.
If it wasn’t isolated, you would rearrange the equation to make it isolated. For example, if you had \((4x - 3)^2 + 2 = 11\), you would first subtract 2 from both sides to isolate the squared term: \((4x - 3)^2 = 9\).

Positive and Negative Roots

When solving a quadratic equation using the square root property, you must include both positive and negative roots. This is because when you square a number, both positive and negative values give the same result.
In the problem \((4x - 3) = \pm 3\), we break it down into:

• \(4x - 3 = 3\)
• \(4x - 3 = -3\)

Solving these will give you:

• For \4x - 3 = 3\: Adding 3 to both sides: \(4x = 6\).
  Dividing by 4: \(x = \frac{3}{2}\).
• For \4x - 3 = -3\: Adding 3 to both sides: \(4x = 0\).
  Thus, \(x = 0\).

So, the solutions are \x = \frac{3}{2}\ and \x = 0\.