Problem 63 Find all solutions of the equati... [FREE SOLUTION]

Chapter 1: Problem 63

Find all solutions of the equation, and express them in the form $a+b i$ $$ x^{2}+x+1=0 $$

Short Answer

Expert verified

The solutions are $ x = -\frac{1}{2} + \frac{i\sqrt{3}}{2} $ and $ x = -\frac{1}{2} - \frac{i\sqrt{3}}{2} $.

Step by step solution

Recognize the Equation Form

The given equation is a quadratic equation of the form $ ax^2 + bx + c = 0 $. In this case, we have $ a=1, b=1, $ and $ c=1 $. This means we can apply the quadratic formula to find the solutions.

Apply the Quadratic Formula

The quadratic formula is given by $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. Substitute $ a = 1 $, $ b = 1 $, and $ c = 1 $ into the formula: \[ x = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times 1}}{2 \times 1} \] This simplifies to: \[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} \] \[ x = \frac{-1 \pm \sqrt{-3}}{2} \]

Simplify the Square Root of a Negative Number

The term $ \sqrt{-3} $ can be expressed as an imaginary number: $ \sqrt{-3} = i\sqrt{3} $. Substitute $ \sqrt{-3} = i\sqrt{3} $ into the expression: \[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

Express in form $ a + bi $

Now, simplify the results: \[ x = \frac{-1}{2} \pm \frac{i\sqrt{3}}{2} \] This gives us the two solutions as: $ x = -\frac{1}{2} + \frac{i\sqrt{3}}{2} $ and $ x = -\frac{1}{2} - \frac{i\sqrt{3}}{2} $. These solutions are already in the form $ a + bi $.

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Key Concepts

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

Imaginary Numbers

Imaginary numbers might seem a bit puzzling at first, but they are easier to understand with a little practice. They are really just numbers that involve the square root of negative numbers. Usually, you cannot take the square root of a negative number within the realm of real numbers. That鈥檚 where imaginary numbers come into play.
When you see an expression like $ \sqrt{-3} $, it can be rewritten using the imaginary unit $ i $, where $ i = \sqrt{-1} $. So, $ \sqrt{-3} $ becomes $ i\sqrt{3} $.

The concept of imaginary numbers extends our understanding of quantity and mathematical reality.
They are a critical part of complex numbers, which combine both real and imaginary numbers.

In the problem you鈥檙e working on, recognizing that $ \sqrt{-3} = i\sqrt{3} $ is key to expressing your solutions.

Quadratic Formula

The quadratic formula is a reliable tool to solve any quadratic equation, no matter how complex. Standard quadratic equations have a form like $ ax^2 + bx + c = 0 $. The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula is very powerful because it gives solutions to any quadratic equation by converting the complex equation into a straightforward calculation.

The expression under the square root $ b^2 - 4ac $ is called the discriminant.
If the discriminant is positive, the equation has two distinct real solutions.
If it is zero, there is one real solution.
If it is negative, the solutions will be complex or imaginary.

In the given problem, the discriminant $ 1 - 4 = -3 $ shows that imaginary numbers are involved, leading us to the solution $ x = \frac{-1 \pm i\sqrt{3}}{2} $. Knowing how to use and apply the formula is essential for solving such equations successfully.

Complex Numbers

Complex numbers are a crucial concept in mathematics, particularly useful for solving equations where imaginary numbers are involved. They take the form $ a + bi $, where $ a $ is the real part and $ bi $ represents the imaginary part. Complex numbers come in handy in various fields, including engineering and physics.

Every real number can be considered a complex number with an imaginary part of zero.
When you add or subtract complex numbers, you deal with the real and imaginary parts separately.
Multiplication and division of complex numbers follow specific rules involving real and imaginary components.

In the context of the problem you鈥檙e working on, the solutions $ x = -\frac{1}{2} + \frac{i\sqrt{3}}{2} $ and $ x = -\frac{1}{2} - \frac{i\sqrt{3}}{2} $ are presented in the form $ a + bi $. This form is standard for expressing complex solutions, ensuring that both the real and imaginary components are clear.

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91影视

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}+x+1=0 $$

Short Answer

Step by step solution

Recognize the Equation Form

Apply the Quadratic Formula

Simplify the Square Root of a Negative Number

Express in form \( a + bi \)

Key Concepts

Imaginary Numbers

Quadratic Formula

Complex Numbers

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