Problem 23 - Elementary Algebra

Question

Solve. $$ (x-4)(x-4)=0 $$

Step-by-Step Solution

Answer

x = 4.

Step 1: Write the equation

The original equation given is: (x - 4)(x - 4) = 0.

Step 2: Simplify the equation

Recognize that (x - 4)(x - 4) is the same as (x - 4)^2. Rewrite the equation as: (x - 4)^2 = 0.

Step 3: Solve for x

To solve (x - 4)^2 = 0, take the square root of both sides of the equation. This gives: x - 4 = 0.

Step 4: Isolate x

Add 4 to both sides to isolate x: x = 4.

Key Concepts

Factoring Quadratics

In algebra, factoring quadratics is a method used to solve quadratic equations of the form ax^2 + bx + c = 0. Factoring involves breaking down the quadratic equation into the product of two binomials. For example, the given equation (x - 4)(x - 4) = 0 is already factored for us. Each factor represents a possible solution to the equation.

• If (a)(b) = 0, then either a = 0 or b = 0.
• In this case, (x - 4) is both a and b.
• By setting each factor to zero, we get x - 4 = 0.

Factoring is useful when you want to find the roots of a quadratic equation quickly.

Simplifying Equations

Simplifying equations makes them easier to solve. In our example, we start with (x - 4)(x - 4) = 0. We can recognize that (x - 4)(x - 4) is actually (x - 4)^2.

• This shows that we are dealing with a perfect square trinomial.
• We rewrite the equation as (x - 4)^2 = 0.

Rewriting it this way simplifies our process because now we can directly apply the next steps to solve for x.

Isolating Variables

Isolation of variables is crucial for solving equations. To solve (x - 4)^2 = 0, take the square root of both sides. This isolates our variable inside the equation.

• We get x - 4 = 0.
• Our next goal is to have x all by itself on one side of the equation.

To isolate x, you need to undo whatever operations are affecting it. In this case, subtracting 4 from x.

Square Root Method

The square root method is particularly useful for dealing with squared terms like (x - 4)^2. This method involves taking the square root of both sides to simplify the equation.

• For example, (x - 4)^2 = 0 simplifies to x - 4 = 0 when you take the square root of both sides.
• After applying the square root, we solve the resulting linear equation. Add 4 to both sides to isolate x and get x = 4.

This method helps simplify the problem by reducing it from a quadratic to a linear equation.