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Chapter 8: Problem 28

Find all complex-number solutions. $$ \left(t+\frac{3}{2}\right)^{2}=\frac{7}{2} $$

Short Answer

Expert verified
The solutions are \( t = -\frac{3}{2} + \frac{\sqrt{14}}{2} \) and \( t = -\frac{3}{2} - \frac{\sqrt{14}}{2} \).

Step by step solution

01

Isolate the Square Term

First, isolate the square term by taking the square root of both sides of the equation: \[ (t+\frac{3}{2})^2 = \frac{7}{2} \]
02

Apply the Square Root Property

To solve for t, take the square root of both sides of the equation: \[ t+\frac{3}{2} = \pm \frac{\sqrt{7}}{\sqrt{2}} \]Which simplifies to: \[ t+\frac{3}{2} = \frac{\sqrt{14}}{2} \]
03

Isolate t

Now, isolate t by subtracting \(\frac{3}{2}\) from both sides: \[ t = -\frac{3}{2} \pm \frac{\sqrt{14}}{2} \]
04

Simplify the Solution

The expression can be split into two separate solutions: \[ t = -\frac{3}{2} + \frac{\sqrt{14}}{2} \] and \[ t = -\frac{3}{2} -\frac{\sqrt{14}}{2} \]

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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 valuable tool for solving quadratic equations that have been simplified to the form \(x^2 = k\). To apply the square root property, we take the square root of both sides of the equation. This introduces the \(\pm\) symbol, representing both the positive and negative roots of the number. For example, if we have \(x^2 = 9\), taking the square root of both sides gives us \(x = \pm 3\). This method is important in finding real and complex solutions.
Isolating Variable
Isolating the variable is a fundamental step in solving equations. It involves manipulating the equation to get the variable of interest by itself on one side of the equation. Let's take the example from the original exercise: \((t+\frac{3}{2})^2 = \frac{7}{2}\). First, we take the square root of both sides to undo the square, leading to \(t + \frac{3}{2} = \pm \frac{\sqrt{14}}{2}\). Subtracting \frac{3}{2}\ from both sides isolates \(t\), resulting in \(t = -\frac{3}{2} \pm \frac{\sqrt{14}}{2}\). This step is crucial in progressively simplifying the equation.
Solving Quadratic Equations
Solving quadratic equations often requires techniques such as factoring, using the quadratic formula, or completing the square. In our exercise, we used the method of isolating the square term and applying the square root property. Quadratics have the general form \(ax^2 + bx + c = 0\). Solutions can be real or complex, depending on the discriminant \(b^2 - 4ac\). When the discriminant is negative, the solutions are complex and involve imaginary numbers. In this exercise, we dealt with \(t\)'s quadratic form after isolating the square term: \(t = -\frac{3}{2} \pm \frac{\sqrt{14}}{2}\).
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic and manageable form. This is often done by combining like terms, factoring, and reducing fractions. In the context of our exercise, after obtaining \(t = -\frac{3}{2} \pm \frac{\sqrt{14}}{2}\), we split the solution into two parts: \(t = -\frac{3}{2} + \frac{\sqrt{14}}{2}\) and \(t = -\frac{3}{2} - \frac{\sqrt{14}}{2}\). Breaking the equation into its component parts helps in understanding the broad and fine details of the solution. This final step ensures that the solutions are clear and can be interpreted correctly.

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