Problem 51 - Intermediate Algebra

Question

Use Property \(6.1\) to help solve each quadratic equation. $$ (2 x-3)^{2}=1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2\) and \(x = 1\).

Step 1: Understand Property 6.1

Property 6.1 relates to solving quadratic equations by taking square roots. It states that if you have an equation of the form \((a)^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\).

Step 2: Set the Equation According to Property 6.1

Given the equation \((2x - 3)^2 = 1\), identify it as a perfect square in the form \((a)^2 = b\) where \(a = 2x - 3\) and \(b = 1\).

Step 3: Apply the Square Root to Both Sides

Take the square root of both sides in accordance with Property 6.1. This gives two equations: \(2x - 3 = \sqrt{1}\) and \(2x - 3 = -\sqrt{1}\).

Step 4: Solve the First Equation

For the equation \(2x - 3 = 1\), add 3 to both sides to get \(2x = 4\). Then, divide both sides by 2 to find \(x = 2\).

Step 5: Solve the Second Equation

For the equation \(2x - 3 = -1\), add 3 to both sides to get \(2x = 2\). Then, divide both sides by 2 to find \(x = 1\).

Step 6: Verify the Solutions

Substitute \(x = 2\) and \(x = 1\) back into the original equation to ensure both values satisfy it. For both values, \((2x - 3)^2\) equals 1, confirming both are solutions.

Key Concepts

Perfect Squares

In mathematics, a perfect square is a number that can be expressed as the product of an integer with itself. For example, the numbers 1, 4, 9, 16, and 25 are all perfect squares because they can be written as

• 1 = 1 × 1
• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5

This concept extends to algebra, where an expression is a perfect square if it is written as the square of a linear binomial, such as \((a + b)^2 = a^2 + 2ab + b^2\).
When solving quadratic equations, identifying perfect square forms helps simplify the process by allowing the use of square roots to find solutions.
In the context of the given problem, \((2x - 3)^2\) is recognized as a perfect square, as it fits the form \((a)^2=b\).
Recognizing such forms can simplify solving equations by connecting algebraic expressions with numerical computations.

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 16 is 4 because \(4 \times 4 = 16\).
Square roots are crucial for solving equations that involve perfect squares.
The square root symbol is denoted \(\sqrt{}\).
Mathematically, any non-negative number has two square roots: one positive and one negative. Therefore, \(\sqrt{16} = 4\) and \(-\sqrt{16} = -4\).
For quadratic equations in the form \((a)^2 = b\), solving for \(a\) involves taking the square root of \(b\).
In our original equation, \((2x - 3)^2 = 1\), we solve by taking the square root of both sides, implying \(2x - 3 = \sqrt{1}\) and \(2x - 3 = -\sqrt{1}\).
Understanding when and why to take square roots is a key skill in solving quadratic equations efficiently.

Solving Equations

Solving equations is about finding the values of variables that make the equation true. In algebra, this involves doing operations to isolate the variable of interest.
For quadratic equations, particularly ones that are in a perfect square form, we make use of square roots to simplify the process.
The problem \((2x - 3)^2 = 1\) illustrates this principle well. As this is already a perfect square, we take the square root of both sides to break it down into two simpler linear equations:

• \(2x - 3 = 1\)
• \(2x - 3 = -1\)

Then, solve each equation separately by isolating \(x\).
For \(2x - 3 = 1\):

• Add 3 to both sides: \(2x = 4\)
• Divide by 2: \(x = 2\)

For \(2x - 3 = -1\):

• Add 3 to both sides: \(2x = 2\)
• Divide by 2: \(x = 1\)

After finding the solutions, always verify them by plugging them back into the original equation to ensure they satisfy \((2x - 3)^2=1\). This process ensures the solutions are correct and often helps catch any computational errors made during problem-solving.