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Chapter 1: Problem 96

Solve the equation by using any method. \(11 t^{2}+3=0\)

Short Answer

Expert verified
[ t = \pm \frac{i \sqrt{33}}{11} ]

Step by step solution

01

- Subtract 3 from Both Sides

Start by isolating the term with the variable. Subtract 3 from both sides of the equation: \[ 11t^2 + 3 - 3 = 0 - 3 \] This simplifies to: \[ 11t^2 = -3 \]
02

- Divide by 11

Next, divide both sides of the equation by 11 to solve for \( t^2 \): \[ \frac{11t^2}{11} = \frac{-3}{11} \] Simplifying gives: \[ t^2 = -\frac{3}{11} \]
03

- Take the Square Root of Both Sides

Take the square root of both sides to solve for \( t \). Remember that taking the square root of a negative number introduces the imaginary unit \( i \): \[ t = \pm \sqrt{-\frac{3}{11}} \] Simplifying further gives: \[ t = \pm i \sqrt{\frac{3}{11}} \]
04

- Express in Simplest Form

Write the expression in simplest form. Combine the imaginary unit and the square root: \[ t = \pm \frac{i \sqrt{3}}{\sqrt{11}} \] Rationalize the denominator, if necessary: \[ t = \pm \frac{i \sqrt{33}}{11} \]

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Key Concepts

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

imaginary numbers
Imaginary numbers arise when we take the square root of a negative number. Typically, the square root of a negative number is not defined in the set of real numbers. However, with imaginary numbers, we introduce a way to work with these square roots. The unit imaginary number is denoted by the symbol \( i \), which is defined as \( i = \sqrt{-1} \). So, when we see the square root of a negative number, such as \( \sqrt{-3} \), we can express it as \( i \sqrt{3} \). This concept helps us to solve equations that have no real solutions. For instance, in our example, when we take the square root of \( -\frac{3}{11} \), we introduce \( i \) to handle the negative part:
square roots
Understanding square roots is vital for solving quadratic equations. The square root of a number \( n \), denoted as \( \sqrt{n} \), is a value \( x \) such that \( x^2 = n \). When solving quadratic equations, taking the square root of both sides is an essential step. It's important to remember that every non-zero number has two square roots: a positive and a negative. This is why we see the \( \pm \) sign in our final answer:
rationalizing the denominator
Rationalizing the denominator is a process used to eliminate square roots from the denominator of a fraction. It's often useful for creating a more standard or 'nicer' form of the fraction. To rationalize the denominator, we multiply the numerator and the denominator by the square root found in the denominator. In our example:

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