Problem 110 - Contemporary Precalculus

Question

Explain why the given statement is true for any numbers \(c\) and \(d .\) I Hint: Look at the properties of absolute value on page 10.1 $$\sqrt{9 c^{2}-18 c d+9 d^{2}}=3|c-d|$$

Step-by-Step Solution

Answer

Question: Prove the following statement for any numbers c and d: \(|3c - 3d| = \sqrt{9c^2 - 18cd + 9d^2}\). Solution: By following the steps outlined above, we demonstrated that the given statement is true for any numbers c and d. The left side of the equation, \(|3c - 3d|\), simplifies to \(3|c - d|\), which is equal to the right side of the equation, thus proving the statement.

Step 1: Expand the expression under the square root

Start by expanding the expression in the square root: $$9c^2 - 18cd + 9d^2 = (3c-3d)^2$$ Notice that we factored this expression as a perfect square: \((3c-3d)^2\).

Step 2: Take the square root of both sides

Now take the square root of both sides: $$\sqrt{(3c-3d)^2} = \sqrt{9c^2 - 18cd + 9d^2}$$ Since the square root and the square functions are inverse operations, the left side simplifies as follows: $$3c - 3d = \sqrt{9c^2 - 18cd + 9d^2}$$

Step 3: Calculate the absolute value of the difference

We are now interested in the absolute value of the left side. Let's calculate the absolute value of the difference \(3c - 3d\): $$|3c - 3d| = 3|c - d|$$ Here we have factored out the constant 3 from the absolute value, as it is always positive, and it doesn't affect the absolute value of the expression.

Step 4: Final comparison

Finally, we'll compare our result to the right side of the given equation: $$3|c - d| = 3|c - d|$$ The LS and the RS are equal, so we have proven that for any numbers \(c\) and \(d\), the given statement holds true.

Key Concepts

Factoring Perfect Squares

Factoring perfect squares is a useful technique in algebra that simplifies expressions. It's like reverse engineering an expression back into its multiplied form. In our example, we started with the expression \(9c^2 - 18cd + 9d^2\). By noticing the pattern, we factored it into \((3c - 3d)^2\). To do this:

• Recognize patterns that resemble \((a - b)^2 \equiv a^2 - 2ab + b^2\).
• In this case, \(9c^2 = (3c)^2\), \(9d^2 = (3d)^2\), and \(-18cd = -2 \times 3c \times 3d\).

Thus, we factor it, simplifying the process of solving or transforming the expression further.

Inverse Operations

Inverse operations are mathematical operations that reverse the effect of each other. The square and square root functions are perfect examples. When we have a squared expression, the square root operation brings it back to its original base. In the solution:

• We started with \((3c-3d)^2\) and applied the square root to both sides.
• Thus, \(\sqrt{(3c-3d)^2} = 3c-3d\).

This simplification harnesses the power of inverse operations, allowing calculations to become more straightforward.

Square Root Properties

Square root properties are key to understanding how expressions simplify. They tell us that the square root of a squared number is essentially the absolute value of the original number or expression. This holds because:

• For any expression \(x\), \(\sqrt{x^2} = |x|\).
• In our case, after factoring, \(\sqrt{9c^2 - 18cd + 9d^2} = |3c - 3d|\).

This property can handle negative values within squares, bringing them back to positive values through absolute value.

Expression Simplification

Expression simplification allows us to make complex mathematical expressions more manageable. After using properties of square roots and absolute values, we simplified to find:

• \(|3c-3d| = 3|c-d|\).
• This is achieved by factoring out the constant 3, as it doesn’t affect the absolute value.

Simplifying expressions helps us verify or solve equations easily, reducing potential errors in calculation.