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Chapter 8: Problem 36

Find all complex-number solutions. Let \(f(t)=(t+6)^{2} .\) Find \(t\) such that \(f(t)=15\)

Short Answer

Expert verified
The solutions are \( t = -6 + \sqrt{15} \) and \( t = -6 - \sqrt{15} \).

Step by step solution

01

Set the function equal to 15

Start by setting the function equal to 15, since we're given that \( f(t) = (t + 6)^2 \) We need to find \( t \) such that \( (t + 6)^2 = 15 \)
02

Solve for \( t + 6 \)

To isolate \( t \), first take the square root of both sides of the equation: \( \sqrt{(t + 6)^2} = \pm \sqrt{15} \) This simplifies to \( t + 6 = \pm \sqrt{15} \).
03

Isolate \( t \)

Now solve for \( t \) by subtracting 6 from both sides: \( t = -6 + \sqrt{15} \) and \( t = -6 - \sqrt{15} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex numbers
Complex numbers are an essential part of higher mathematics. They include a real part and an imaginary part. The standard form is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In this exercise, although the solutions are not in the form of complex numbers, they are still relevant when solving more advanced equations.
  • The real number \(a\) can be any real value.
  • The imaginary unit \(i\) represents the square root of -1.
  • Complex numbers are used in various fields, including engineering, physics, and applied sciences.
Understanding how complex numbers work will help you as you dive deeper into topics like quadratic equations and roots.
functions
In mathematics, functions are rules that relate inputs to outputs. Essentially, a function takes an input, applies a specific operation to it, and produces an output. In our exercise, the function is given by \(f(t)=(t+6)^2\). This is a way to express how each input value \(t\) maps to an output value of the function.
  • The expression inside the function can change the input value.
  • Squaring the expression ensures the output is always positive (or zero).
  • Functions can be linear, quadratic, exponential, etc., depending on their expressions.
Key things to remember with functions include their domain (all possible input values) and range (all possible output values). Grasping this concept lays the foundation for tackling more complex functions and equations.
square roots
The square root of a number is a value that, when multiplied by itself, gives the original number. This exercise involves taking the square root of both sides of the equation to isolate the variable \(t\). Here are some key points about square roots:
\[ \text{ If } x^2 = N, \text{ then } x = \pm \sqrt{N} \]
  • Square roots can be positive or negative.
  • Every positive number has two square roots: one positive and one negative.
  • The square root of zero is zero, which is why it doesn't have a positive/negative form.
In our context, after isolating \(t + 6\), we took the square root, leading to \(t + 6 = \pm \sqrt{15}\). Remember this approach, as it is crucial for solving quadratic equations.

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