Problem 28 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (x-8)^{2}=0 $$

Step-by-Step Solution

Answer

Answer: The solution to the given quadratic equation is \(x=8\).

Step 1: Identify the equation

In this case, we are given the quadratic equation \((x-8)^2=0\).

Step 2: Apply the square root property

Since the equation is in the form of \((x-8)^2=0\), we can use the square root property to simplify the equation. The square root property states that if \(\alpha^2 = \beta\), then \(\alpha = \pm\sqrt{\beta}\). Apply this property to the given equation: $$ x-8 = \pm\sqrt{0} $$

Step 3: Solve for x

Since the square root of 0 is 0, we have: $$ x-8 = \pm0 $$ Adding 8 to both sides, we get the value of x: $$ x=8 $$ So, the solution to the given equation is \(x=8\).

Key Concepts

Square Root Property

The Square Root Property is an essential tool when solving quadratic equations, especially when the equation is perfectly squared. This property tells us that if you have an equation like \( \alpha^2 = \beta \), you can express \( \alpha \) as \( \pm \sqrt{\beta} \). This means \( \alpha \) can be the positive or negative square root of \( \beta \).
In our equation \((x-8)^2=0\), the square root property helps us directly conclude \(x-8 = \pm\sqrt{0}\). Since the square root of zero is zero, this simplifies to \(x-8 = 0\), indicating a straightforward solution.
This property simplifies the process by breaking down complex quadratic forms into linear equations, which are much easier to solve.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. In the case of a quadratic equation such as \((x-8)^2 = 0\), the objective is to determine the value of \(x\) that satisfies the equation.
This can be achieved by systematically isolating \(x\) using algebraic operations:

• Identify the structure and form of the equation.
• In our case, apply the Square Root Property to simplify the equation to \(x-8 = 0\).
• Solve for \(x\) by isolating it through addition, subtraction, multiplication, or division as needed.

For our example, adding \(8\) to both sides gives us \(x = 8\), which solves the equation.

Algebraic Identities

Algebraic identities are fundamental truths in mathematics that simplify the process of solving equations. They describe predictable patterns or properties in numbers and expressions. One common identity utilized is the perfect square identity: \((a-b)^2 = a^2 - 2ab + b^2\).
Understanding these identities allows us to recognize forms like \((x-8)^2\) and apply the Square Root Property efficiently because we know the expansion leads to easy solutions when set to zero.
By acknowledging these identities, you can:

• Predict the steps needed to simplify and solve equations.
• Understand the underlying structure of quadratic equations.
• Quickly assess which algebraic tools to apply, such as factoring or using the square root property.

These identities are the building blocks that enhance our ability to maneuver through and solve equations with confidence.