Problem 64 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ 2 x^{2}+x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-1 + i\sqrt{7}}{4}\) and \(x = \frac{-1 - i\sqrt{7}}{4}\).

Step 1: Identify the form of the quadratic equation

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

Step 2: Use the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting \(a = 2\), \(b = 1\), and \(c = 1\), we have:\[x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}\].

Step 3: Calculate the discriminant

The discriminant \(\Delta\) is \(b^2 - 4ac\). So, \(\Delta = 1^2 - 4 \cdot 2 \cdot 1 = 1 - 8 = -7\). The discriminant is negative, indicating complex roots.

Step 4: Simplify the expression

Since the discriminant is negative, the solutions will involve imaginary numbers. Substitute back into the quadratic formula:\[x = \frac{-1 \pm \sqrt{-7}}{4}\].

Step 5: Express the solutions in standard form

To express the solutions with imaginary numbers, rewrite \(\sqrt{-7}\) as \(i\sqrt{7}\), where \(i\) is the imaginary unit:\[x = \frac{-1 \pm i\sqrt{7}}{4}\].The solutions are thus \(x = \frac{-1 + i\sqrt{7}}{4}\) and \(x = \frac{-1 - i\sqrt{7}}{4}\).

Key Concepts

Complex Numbers

In algebra, **complex numbers** play a crucial role when dealing with equations where the solutions are not real numbers. A complex number has the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) representing the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

The reason complex numbers are needed in the context of solving quadratic equations is that some quadratic equations do not have real number solutions. This happens when the discriminant (see the section on discriminant) is negative. A negative discriminant implies that the square root of a negative number needs to be calculated, which introduces the imaginary unit \(i\).

• Imaginary Unit \(i\): Satisfies \(i^2 = -1\)
• Complex Number Form: \(a + bi\)
• Example: \(-1 + i\sqrt{7}\)

Learning about complex numbers opens up the complete spectrum of number solutions extending beyond real numbers, making it possible to solve any quadratic equation mathematically.

Quadratic Formula

The **quadratic formula** is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula provides a direct way to find the roots of these equations without needing to factor or complete the square manually.

The formula is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Let's break down the parts:

• \(-b\): This term changes the sign of \(b\).
• \(\pm\): Indicates there are two possible solutions, one with addition and one with subtraction.
• \(\sqrt{b^2 - 4ac}\): Known as the discriminant, it determines the nature of the roots.
• \(2a\): Is the denominator that ensures correct scaling of the result.

Using this formula, any quadratic equation can be solved swiftly and efficiently if you know the coefficients \(a\), \(b\), and \(c\). Remember, when the discriminant is negative, the solutions will be complex numbers involving the imaginary unit \(i\).

Discriminant

The **discriminant** is a component of the quadratic formula, and it can tell us a lot about the nature of the roots of the quadratic equation we're dealing with. It is represented as \(b^2 - 4ac\) and is found under the square root in the quadratic formula.

Let's explore what the value of the discriminant means for the solutions:

• **Positive Discriminant:** Indicates two distinct real roots. This happens when \(b^2 - 4ac > 0\).
• **Zero Discriminant:** Indicates a perfect square, leading to exactly one real root (a repeated root), occurring when \(b^2 - 4ac = 0\).
• **Negative Discriminant:** Indicates two complex conjugate roots. Here, \(b^2 - 4ac < 0\) implies the roots involve imaginary numbers.

In the equation \(2x^2 + x + 1 = 0\), the discriminant was calculated to be \(-7\). Since it is negative, this tells us ahead of time that our solutions will be complex, involving \(i\), the imaginary unit.