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Chapter 10: Problem 59

Solve the equation by using the Square Root Property. $$(t-3)^{2}=-25$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(t = 3+5i\) and \(t = 3-5i\).

Step by step solution

01

Apply the Square Root Property

Taking the square root of both sides transforms the equation to \(t-3 = 卤sqrt(-25)\).
02

Solve for the imaginary part of the root

As sqrt(-25) is an imaginary number, it can be rewritten as \(5i\) because we know that the square root of -1 is \(i\). Hence, we have \(t-3 = 卤5i\).
03

Solve for t

Finally, we isolate \(t\) by adding 3 to both sides of the equation. This yields \(t = 3 卤5i\).

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Key Concepts

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

Square Root Property
The Square Root Property is a helpful tool when solving quadratic equations in the form \((x - a)^2 = b\). By applying this property, you can effectively simplify an equation by taking the square root of both sides.
  • This process effectively "undoes" the squaring operation.
  • It's important to remember that when you take the square root, you must consider both the positive and negative roots. This is denoted as \(卤\sqrt{b}\).
In context, the equation \((t-3)^2 = -25\) becomes \(t-3 = 卤\sqrt{-25}\) as soon as you apply the Square Root Property.
This step is crucial in transitioning from a potentially complex quadratic equation into a simpler linear form.
Imaginary Numbers
Imaginary numbers come into play when you take the square root of a negative number. The most basic imaginary number is \(i\), which is defined as the square root of -1.
  • Imaginary numbers are used to extend the real number system.
  • They allow for solutions to equations that would otherwise have no real answer.
In our problem, after applying the Square Root Property, we are left with \(\sqrt{-25}\), which can be rewritten as \(5i\).
This transformation is achieved by recognizing that \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1}\), simplifying to \(5 \cdot i\).
Imaginary numbers therefore enable us to progress further in solving equations rooted in negative terms.
Complex Solutions
Complex solutions are a natural extension of working with imaginary numbers. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
  • In the context of quadratic equations, if the discriminant (the part under the square root in the quadratic formula) is negative, the solutions will be complex.
  • This is due to the involvement of an imaginary number, as seen in our equation result: \(t = 3 \pm 5i\).
These solutions use both real and imaginary parts, reflecting a complete set of all possible outcomes.
Complex solutions are not just abstract concepts; they play vital roles in fields like engineering and physics, where they model real-world phenomena.
Isolating Variables
Isolating the variable is an essential skill in algebraic manipulation. It involves rearranging an equation to solve for the unknown.
  • It often requires adding, subtracting, multiplying, or dividing both sides of an equation.
  • Each operation is aimed at "isolating" the variable on one side.
In our problem, we needed to isolate \(t\) by getting rid of the terms around it. After dealing with the imaginary roots, we added 3 to each side of \(t-3 = \pm 5i\).
This crucial step gives the final solutions: \(t = 3 + 5i\) and \(t = 3 - 5i\).
Mastering the isolation of variables is fundamental in finding solutions across numerous mathematical scenarios.

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