Problem 63 - Precalculus Mathematics for Calculus

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

Step 1: Identify the Quadratic Equation Form

The given equation is \( x^2 + x + 1 = 0 \). This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 1 \), and \( c = 1 \).

Step 2: Use the Quadratic Formula

The quadratic formula to find the roots of the equation \( ax^2 + bx + c = 0 \) is given by \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). Substitute \( a = 1 \), \( b = 1 \), and \( c = 1 \) into the formula.

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is \( b^2 - 4ac = 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3 \). Since the discriminant is negative, the solutions will be complex numbers.

Step 4: Find the Complex Solutions

Substitute the discriminant into the quadratic formula: \( x = \frac{-1 \pm \sqrt{-3}}{2} \). This can be expressed as \( x = \frac{-1 \pm i\sqrt{3}}{2} \), since \( \sqrt{-3} = i\sqrt{3} \).

Step 5: Express the Solutions in the Required Form

The solutions are \( x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \) and \( x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \). These are expressed in the form \( a+bi \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The term "quadratic" comes from the Latin word for "square," indicating the highest degree of the variable is squared. Quadratic equations are fundamental in algebra and arise in various physical, financial, and mathematical contexts.

To identify a quadratic equation, look for three distinct terms: a term with \( x^2 \), a term with \( x \), and a constant term. In the case of the equation \( x^2 + x + 1 = 0 \), the coefficients are \( a = 1 \), \( b = 1 \), and \( c = 1 \). This equation is perfectly set for solving using methods like factoring, completing the square, or using the quadratic formula.

Quadratic equations can have real or complex roots, depending on the values of their coefficients and the discriminant. Understanding what type of solutions a quadratic equation might have is crucial for solving it correctly.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation \( ax^2 + bx + c = 0 \). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula gives two solutions because of the "\( \pm \)" sign, which indicates the roots are found by adding and subtracting the value of the square root term.

To use the quadratic formula effectively, substitute the values of \( a \), \( b \), and \( c \) from your equation into this formula. In the example \( x^2 + x + 1 = 0 \), we substitute \( a = 1 \), \( b = 1 \), and \( c = 1 \). The formula then becomes:\[x = \frac{-1 \pm \sqrt{1 - 4 \times 1 \times 1}}{2 \times 1} = \frac{-1 \pm \sqrt{-3}}{2}.\]

The quadratic formula can handle any type of quadratic equation, providing either real or complex solutions, depending on the discriminant. It's an essential method for solving quadratic equations, especially when other techniques are not feasible.

Discriminant

The discriminant in a quadratic equation \( ax^2 + bx + c = 0 \) is represented by the expression \( b^2 - 4ac \). This value is crucial because it tells us the nature of the roots of the quadratic equation.

Here's how the discriminant works:

• If \( b^2 - 4ac \gt 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), it has exactly one real root (or a repeated root).
• If \( b^2 - 4ac \lt 0 \), the roots are complex conjugates, meaning they have real and imaginary components.

In the exercise \( x^2 + x + 1 = 0 \), the discriminant calculated is \(-3\) (that is \( 1 - 4 \cdot 1 \cdot 1 \)). Since the discriminant is negative, the solutions of the quadratic equation are complex numbers and can be expressed in the form \( a + bi \).

The discriminant is an indicator of the equation's complexity and helps us pre-determine the method to use for solving the quadratic equation, depending on whether we expect real or complex solutions.