Problem 73 - Elementary Algebra

Question

Solve using the quadratic formula. $$ 2 x_{2}+3 x+2=0 $$

Step-by-Step Solution

Answer

The roots are complex: \( x = \frac{-3 + i\sqrt{7}}{4} \) and \( x = \frac{-3 - i\sqrt{7}}{4} \).

Step 1: Identify the Quadratic Equation

The given equation is \( 2x^2 + 3x + 2 = 0 \), which is already in the standard quadratic form \( ax^2 + bx + c = 0 \). Here, \( a = 2 \), \( b = 3 \), and \( c = 2 \).

Step 2: Recall the Quadratic Formula

The quadratic formula to find the roots of \( ax^2 + bx + c = 0 \) is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. We will use this formula to solve the equation.

Step 3: Calculate the Discriminant

The discriminant is given by \( b^2 - 4ac \). Substitute the values: \( b = 3 \), \( a = 2 \), and \( c = 2 \). Thus, the discriminant is \( 3^2 - 4 \times 2 \times 2 = 9 - 16 = -7 \).

Step 4: Apply the Quadratic Formula

Since the discriminant is negative, the equation has complex roots. Apply the formula: \[ x = \frac{-3 \pm \sqrt{-7}}{4} \]. Simplify to find the roots. The solutions are \[ x = \frac{-3 + i\sqrt{7}}{4} \] and \[ x = \frac{-3 - i\sqrt{7}}{4} \], where \( i \) is the imaginary unit.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. This means its highest exponent on the variable, typically denoted as *x*, is two. These types of equations are generally presented in the form:

• \( ax^2 + bx + c = 0 \)

Here, *a*, *b*, and *c* are constants, with *a* not equal to zero, because if *a* were zero, the equation would not be quadratic but linear. Quadratic equations can have different forms of solutions, depending on the value of the discriminant. They can have two real roots, one real root, or no real roots, instead having two complex roots. These equations are fundamental in algebra, and their solutions provide intersections of the parabola \( y = ax^2 + bx + c \) with the x-axis.

Discriminant

The discriminant of a quadratic equation is crucial because it tells us about the nature of the roots without actually solving the equation. It is found using the formula:

• \( b^2 - 4ac \)

In our example: \( b = 3 \), \( a = 2 \), \( c = 2 \) lead to a discriminant of \( 3^2 - 4 \times 2 \times 2 \), simplifying to \( 9 - 16 \) which equals \(-7\).
Based on the value of the discriminant, we determine:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root (known as a double root).
• If \( b^2 - 4ac < 0 \), there are no real roots, but two complex roots.

Complex Roots

Complex roots occur when the discriminant of a quadratic equation is negative, meaning there are no real roots as the parabola does not cross the x-axis. Instead, it implies solutions in the form of complex numbers, which include an imaginary part. Complex numbers are expressed as

• \( a + bi \)

where *i* is the imaginary unit, equal to \(\sqrt{-1}\). For our quadratic equation, the negative discriminant \(-7\) indicates complex roots. Using the quadratic formula, the solutions are

• \( x = \frac{-3 \pm i\sqrt{7}}{4} \).

These solutions indicate that the solutions are not on the real number line but rather in the complex plane, where they are represented as complex conjugates.

Standard Form

The standard form of a quadratic equation is an essential concept because it allows for easy application of the quadratic formula and other algebraic techniques. It is typically written as:

• \( ax^2 + bx + c = 0 \)

In this setup, *a*, *b*, and *c* are known as coefficients, and *a* is not zero. Arranging a quadratic equation in this form helps identify these coefficients quickly, which are crucial for calculating the discriminant and employing the quadratic formula. Our example equation, \( 2x^2 + 3x + 2 = 0 \), perfectly fits this structure, making it ready for immediate analysis using the quadratic formula. The standard form serves as a foundational structure in solving and understanding quadratic relationships.