Problem 60 - 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

Answer: (a) A = 0 (b) A is any positive number (c) There is no negative solution.

Step 1: Solution \(t=0\)#

In order for the solution \(t=0\), we need to find the value of \(A\) for which \(t^4 = A\) is true when \(t=0\). Just substitute test value \(t=0\) into the equation: \[(0)^4 = A\] \[A = 0\] So, for \(A=0\), there is a solution \(t=0\).

Step 2: Positive solution#

We need to find values of \(A\) for which the solution \(t\) is positive. Since \(t^4\) is always non-negative (because even power of any real number is non-negative), if \(A\) is positive, there will always be a positive value of \(t\) such that \(t^4 = A\). Therefore, for any positive value of \(A\), the equation \(t^4 = A\) has a positive solution.

Step 3: Negative solution#

If we want to find values of \(A\) for which the solution \(t\) is negative, since \(t^4\) is always non-negative, there is no value of \(A\) which would make the equation \(t^4 = A\) have a negative solution. In other words, there is no negative solution for this equation.

Key Concepts

Real Solutions

In a quartic equation like \(t^4 = A\), a real solution means that there is a real number \(t\), which when raised to the fourth power, results in \(A\). For the equation given, any value of \(A\) that does not make the equation complex or undefined will lead to real solutions.

• When \(t=0\), the equation simplifies to \((0)^4 = A\), which means \(A=0\). This is a real solution because 0 is a real number.
• If \(A\) is positive, \(t^4 = A\) will also always have real solutions, since positive values can always be a result of real numbers being raised to even powers.
• However, there are no real solutions when \(A\) is negative because an even power of any real number cannot be negative.

Positive Solutions

To have a positive solution for \(t\) in the equation \(t^4 = A\), the value \(A\) must be greater than zero. This is because if \(A\) is positive, there is a positive real number \(t\) such that \(t^4 = A\).

• The key idea is that raising any positive number to an even power will still be positive, which aligns perfectly with a positive value of \(A\).
• For example, if \(A = 16\), then \(t\) could be \(2\), since \(2^4 = 16\).
• As long as \(A\) remains a positive number, the equation \(t^4 = A\) will always yield positive solutions.

Negative Solutions

When considering negative solutions for \(t\) in a quartic equation, it's important to understand the nature of even powers.

• Even though \(t\) itself could potentially be negative, raising it to the fourth power results in a positive value. Hence, \(t^4\) cannot equal a negative value for any real number \(t\).
• This means there cannot be any solution for \(t\) such that \(t^4 = A\) where \(A\) is negative. The equation simply does not work this way for real numbers.
• This is why in the case of \(t^4 = A\), negative values for \(A\) do not yield any solutions that are real. Essentially, a negative solution in the real number domain is impossible.