Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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: \[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} \] \[ x = \frac{-1 \pm \sqrt{-3}}{2} \]

Step 3: 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: \[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

Step 4: 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 \( a + bi \).

Key Concepts

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’s 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’re 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’re 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.