Problem 59 - 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 $$ t^{3}=A $$

Step-by-Step Solution

Answer

Answer: (a) A=0.

Step 1: Part (a) - Solution t = 0

If \(t = 0\), then \(0^3 = 0 = A\). So \(t = 0\) is a solution when \(A = 0\).

Step 2: Part (b) - Positive solution

If \(t > 0\), then \(t^3 > 0\), so \(A > 0\). The equation \(t^3 = A\) has a positive solution when \(A > 0\), namely \(t = \sqrt[3]{A}\).

Step 3: Part (c) - Negative solution

If \(t < 0\), then \(t^3 < 0\), so \(A < 0\). The equation has a negative solution when \(A < 0\), namely \(t = \sqrt[3]{A}\) (which is negative when \(A\) is negative).

Key Concepts

Solutions of Equations

When tackling a cubic equation like \(t^3 = A\), the goal is to find all possible values of \(t\) that satisfy the equation for a given \(A\). Solutions to equations mean finding the roots, or the values of \(t\), which when cubed, equal \(A\). Cubic equations like these are intriguing because they have three potential roots. This means they can have upto three solutions including repeated roots. Unlike quadratic equations which have up to two solutions, cubic equations open up the possibility for more varied solutions. It's essential to understand that a cubic equation, no matter what the value of \(A\) is, will always have at least one real root. This stems from the nature of real numbers and the continuous growth of the cubic function. This real root could be a positive number, a negative number, or even zero. So, whenever you encounter a problem of this nature, remember that your task is to determine the type and nature of solutions based on the constant \(A\).

Positive Solutions

A positive solution to a cubic equation \(t^3 = A\) implies finding the value of \(t\) that is greater than zero. For a real number \(t\) to satisfy \(t^3 = A\) and be positive, the constant \(A\) must itself be positive. Here's why:

• If \( A > 0 \), then \( t^3 = A \) means \( t \) has to also be positive since the cube of a positive number is positive.
• This gives us a straightforward situation where any positive \(A\) assures us of a legitimate positive solution for \(t\).

Thus, if you're looking for a positive value of \(t\), make sure \(A\) is positive. It's a simple yet crucial relation to remember: positive \(A\) leads to a positive real solution for \(t\).

Negative Solutions

A negative solution in the context of the cubic equation \(t^3 = A\) emerges when \(t\) is a negative number. Understanding how negative solutions work involves recognizing how negative numbers behave under cubing.For any \(A\) that is negative:

• If \( A < 0 \), it implies that \( t \) must also be negative because the cube of a negative number is negative.
• This means a negative \(A\) ensures at least one negative solution for \(t\).

The principle to remember is quite straightforward: if you see a negative \(A\), then you have a negative solution for \(t\). This understanding of negative solutions is pivotal, as it uses the inherent properties of numbers to establish the nature of the equation's roots.