Problem 62 - Algebra Form and Function

Question

In Problems \(59-70,\) decide for what values of the constant \(A\) the equation has (a) The solution \(t=0\) (b) A positive solution (c) A negative solution $$ (-t)^{4}=A $$

Step-by-Step Solution

Answer

To summarize, for the equation \((-t)^{4} = A\), the following conditions apply: a) For the solution \(t=0\), the value of \(A\) must be 0. b) For a positive solution, the value of \(A\) must be greater than or equal to 0. c) There are no values of \(A\) that yield a negative solution.

Step 1: General Solution

First, notice that the term \((-t)^4\) will always be non-negative, since any power, even a negative one like \(-t\), of 4 will always be positive (or zero if \(t=0\)). Thus, the equation is defined for all real numbers. Now let's find the general solution: $$ (-t)^{4}=A $$ We can simplify this equation by taking the fourth root of both sides: $$ t = \pm\sqrt[4]{A} $$ **Step 2: Find the values of A for the solution t=0**

Step 2: Solution t=0

Now let's apply each condition. For (a), we want to find the values of \(A\) for which the solution is \(t=0\). Substitute 0 into the general solution, then solve for \(A\): $$ 0 = \pm\sqrt[4]{A} $$ Since the fourth root of any non-zero number will never be 0, the only possible value for \(A\) is: $$ A = 0 $$ **Step 3: Find the values of A for a positive solution**

Step 3: Positive Solution

For (b), we want to find the values of \(A\) for which the solution is positive. Since the general solution is given by \(t = \pm\sqrt[4]{A}\), we can see that any non-negative value for A will have a positive solution. This includes the previously found value \(A=0\) Thus, for a positive solution, the possible values for \(A\) are: $$ A \geq 0 $$ **Step 4: Find the values of A for a negative solution**

Step 4: Negative Solution

For (c), we want to find the values of \(A\) for which the solution is negative. Since \((-t)^4 = A\), the left-hand side is always non-negative and any negative value of \(A\) will not have a real solution, and thus no negative solution. Therefore, there are no possible values for \(A\) that yields a negative solution.

Key Concepts

Non-negative numbers

Non-negative numbers are numbers that are either zero or positive. They are a core concept in algebra and are incredibly important when dealing with equations that include powers or radicals. In the context of the given equation,

• The expression \((-t)^4\) simplifies to always yield a non-negative result because raising any real number to the fourth power will result in a non-negative number, whether you start with negative or positive inputs.
• This means our value of A must be non-negative if the equation is to have a feasible solution with real numbers.
• When you consider non-negative numbers in equations, it influences the kind of solutions you can expect, such as only positive or zero roots.

Non-negative numbers are essential for understanding why some values of A lead to different solutions in our exercise. They limit the solutions based on whether A can allow for zero or positive results.

Fourth roots

The fourth root of a number is what you multiply by itself four times to get that number. It's represented mathematically by the symbol \(\sqrt[4]{\cdot}\). When we take the fourth root of a number:

• We find two potential solutions: a positive and a negative root, as seen in \(t = \pm \sqrt[4]{A}\).
• This is because multiplying a negative number by itself an even number of times (like four) will produce a positive product.
• Understanding the concept of fourth roots helps to find whether there can be a real solution based on the value of A.

In our exercise, this concept is crucial. If you look at the way we solve for \(t\), it involves taking the fourth root of A to determine possible values for t, which limits solutions to non-negative \(A\). Thus, understanding fourth roots is central to solving the exercise correctly.

Real numbers

Real numbers include all rational and irrational numbers, essentially any number that can be found on the number line. They play a significant role here:

• The equation \((-t)^4 = A\) is defined for all real values of \(t\), but only valid for non-negative \(A\). This is due to the nature of raising numbers to even powers.
• In algebra, solving equations for real numbers means ensuring any manipulations or operations conform to the rules of the real number system.
• Real numbers ensure that when taking radicals or dealing with powers, we have numerically viable solutions, being either positive or zero as seen by the solutions to \(t\).

The importance of focusing on real numbers in this exercise is demonstrated by examining what values of A lead to valid solutions or not. It explains why no negative values are possible as solutions for A because they do not yield real, numerically valid outcomes.