Problem 49 - Algebra Form and Function

Question

The equation $$ x \sqrt{8-x}=5 $$ has two solutions. Are they positive, zero, or negative? Give an algebraic reason why this must be the case. You need not find the solutions.

Step-by-Step Solution

Answer

Answer: The solutions must be positive.

Step 1: Observe the Terms

First, let's look at the equation \(x \sqrt{8-x} = 5\). Notice that x is multiplied by \(\sqrt{8-x}\), which means that both of these parts should have the same sign (either positive or negative) in order for their product to be positive. Recall that square roots are always non-negative, meaning \(\sqrt{8-x}\) will be greater than or equal to 0. Also, note that 5 is a positive number on the right side of the equation.

Step 2: Determine the Sign of x

As we know from the previous step, \(\sqrt{8-x}\) must be non-negative. Since the product of x and \(\sqrt{8-x}\) is a positive number (5), x must also be positive to maintain the equation. In other words, since we need the product \(x \sqrt{8-x}\) to be positive, and we know that \(\sqrt{8-x}\) is non-negative, x must be positive as well. A negative value for x would not satisfy the equation, as the product would become negative.

Step 3: Explain Algebraically

We can now provide an algebraic reason for our conclusion. Assume that both the solutions to the given equation are positive. Let's call these solutions \(x_1\) and \(x_2\). So, we have \(x_1 \sqrt{8-x_1} = 5\) and \(x_2 \sqrt{8-x_2} = 5\). As both x and \(\sqrt{8-x}\) are non-negative, and their product is a positive value (5), it holds true that the solutions must be positive.

Step 4: Conclusion

In conclusion, the two solutions of the given equation \(x \sqrt{8-x} = 5\) must be positive values, as the product of x and \(\sqrt{8-x}\) is positive and \(\sqrt{8-x}\) is non-negative.

Key Concepts

Solutions of an Equation

Solutions of an equation refer to the values of the variable that satisfy the equation. In algebra, finding the solutions is like discovering the mystery number that makes the equation true. When we have an equation such as \(x \sqrt{8-x} = 5\), the solutions can be seen as the particular values of \(x\) for which this equation holds.

To determine these solutions, we look for \(x\) values that make both sides of the equation equal. Depending on the form and components of the equation, the solutions can be positive, negative, or even zero. It is important to pay attention to the signs and values assigned to both sides of the equation. This indicates whether particular solutions are valid.

• If the resultant value is defined, the solution is a possible valid answer to the equation.
• However, any indeterminate forms or results that defy the rules of arithmetic may invalidate a solution.

This algebraic approach ensures that we find true solutions, exploring their nature and implications for the given problem.

Positive and Negative Numbers

Understanding positive and negative numbers is essential in tackling algebraic equations. These numbers play a critical role in the sign of the equations and solutions.

• Positive numbers are greater than zero. They are usually denoted without any special symbol preceding them. Examples include 1, 2, and 3.
• Negative numbers are less than zero and are always preceded by a minus sign, such as -1, -2, and -3.

In the context of the equation \(x \sqrt{8-x} = 5\):

• We see that the right side of the equation is a positive number, which is 5.
• Thus, for the equation to hold true, the product on the left side \((x \sqrt{8-x})\) must also produce a positive number.
• This prompts the necessity for our \(x\) to be positive, since the square root term \(\sqrt{8-x}\) is non-negative, meaning it yields zero or positive values only.

This understanding aids in evaluating potential solutions and ensures all calculations adhere to the principles of arithmetic.

Square Roots in Algebra

Square roots are a fundamental concept in algebra that involves determining which number multiplied by itself gives a certain value. The square root of a number \(a\) is denoted as \(\sqrt{a}\).

Square roots have some important properties:

• They are always non-negative for real numbers, meaning the square root of a positive number is positive, and the square root of zero is zero.
• A square root cannot be negative since multiplying two positive numbers or two negative numbers cannot result in a negative product.

Applying this to our equation \(x \sqrt{8-x} = 5\):

• The term \(\sqrt{8-x}\) signifies that the expression \(8-x\) should be non-negative.
• This means \(x\) must be less than or equal to 8 for the square root to be valid.
• Recognizing this helps us determine the range of valid \(x\) values while solving the equation.

Understanding square roots enables us to analyze equations accurately and ensures that we adhere to the correct rules while solving algebraic problems.