Problem 51 - Elementary Algebra

Question

Solve using the Square Root Property. \((x-6)^{2}+7=3\)

Step-by-Step Solution

Answer

There are no real solutions.

Step 1 - Isolate the squared term

Start by isolating the squared term \( (x-6)^{2} \). To do this, subtract 7 from both sides of the equation. \[ (x-6)^{2} + 7 - 7 = 3 - 7 \]This simplifies to \( (x-6)^{2} = -4 \).

Step 2 - Recognize that there is no real solution

The equation \( (x-6)^{2} = -4 \) implies that a squared term is equal to a negative number. Since the square of any real number is non-negative, a squared term cannot be equal to a negative number. Therefore, there are no real solutions to the equation.

Key Concepts

Isolating Squared Terms

To solve equations using the Square Root Property, the first crucial step is isolating the squared term. In the exercise provided, we start with the equation (x-6)^2 + 7 = 3 . The term (x-6)^2 is our squared term that needs to be isolated. We achieve this by subtracting 7 from both sides of the equation. The subtraction looks like this:
(x-6)^2 + 7 - 7 = 3 - 7 . Simplifying, we get (x-6)^2 = -4 .

• Importance: Isolating the squared term makes it easier to apply the Square Root Property.
• Key Tip: Always perform the same operation on both sides of the equation to maintain equality.

Once the squared term is isolated, we can move to further steps in solving the equation.

Real Solutions

In the context of solving equations, a solution is considered 'real' if it is a real number. Real numbers include all the numbers on the number line, including both positive and negative numbers, as well as zero.

When we isolate the squared term in our exercise, we get (x-6)^2 = -4 . The next step would typically be to take the square root of both sides. However, the result we see here is -4 , a negative number.

• Important Note: The square of any real number (whether positive or negative) is always non-negative.
• Key Insight: Since no real number exists that can be squared to give a negative result, there are no real solutions to the equation.

The critical takeaway here is the nature of real solutions. Whenever a squared term equals a negative number, it indicates that the equation does not have a real solution.

Negative Numbers

Negative numbers are numbers that are less than zero. These numbers have important properties and implications in mathematics.

In the provided exercise, when we isolate the squared term, we reach (x-6)^2 = -4 . The presence of the negative number -4 on the right side is a signal to pause and reflect on the properties of squared terms.

• Squaring properties: The square of any real number, whether large or small, positive or negative, can never result in a negative number.
• Understanding zero: Zero is the boundary between negative and positive numbers. Any number squared stops at zero, meaning it cannot become less than zero.

Consequently, an equation like (x-6)^2 = -4 implies an impossible situation under real numbers. Therefore, recognizing and understanding the concept of negative numbers is crucial in solving and analyzing such equations.