Problem 34 - Algebra 2

Question

Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary. $$ 2 x^{2}+x+28=0 $$

Step-by-Step Solution

Answer

The discriminant \(\Delta\) is \(-223\), indicating the quadratic equation has two imaginary solutions.

Step 1: Understand the quadratic formula and the discriminant

The standard form of a quadratic equation is given by ax^2+bx+c = 0. The discriminant of a quadratic equation, represented by the symbol \(\Delta\), is found by the formula \(\Delta = b^2 - 4ac\). The discriminant can tell us the nature and the number of solutions of the quadratic equation: if \(\Delta > 0\), the equation has two real and distinct solutions; if \(\Delta = 0\), the equation has exactly one real solution (a repeated root); and if \(\Delta < 0\), the equation has two imaginary solutions.

Step 2: Identify a, b, and c from the equation

From the given quadratic equation \(\2x^{2} + x + 28 = 0\), identify the coefficients: \(\ a = 2\), \(\b = 1\), and \(\c = 28\).

Step 3: Compute the discriminant

Substitute the values of a, b, and c into the discriminant formula to get \(\Delta = (1)^{2} - 4 \cdot (2) \cdot (28)\), which simplifies to \(\Delta = 1 - 224 = -223\).

Step 4: Determine the number and type of solutions

Since \(\Delta < 0\), the quadratic equation \(\2x^{2} + x + 28 = 0\) has two imaginary solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x, with a nonzero coefficient for the squared term. It has the standard form of \(ax^{2} + bx + c = 0\), where \(a\) is the coefficient of \(x^{2}\), \(b\) is the coefficient of \(x\), and \(c\) represents the constant term, with the stipulation that \(a eq 0\).

The solutions to this type of equation are the values of \(x\) that make the equation true, and these solutions are also termed as 'roots' of the quadratic equation.

Every quadratic equation is guaranteed to have two solutions, which may be real, imaginary, or one solution counted twice, depending on the nature of the discriminant which will be discussed in the following sections.

• A positive discriminant indicates two distinct real solutions.
• A zero discriminant indicates exactly one real solution.
• A negative discriminant indicates two complex solutions.

Imaginary Solutions

Imaginary solutions arise in quadratic equations when the discriminant, \( \(\Delta\) \), is less than zero. This negative value indicates that the square root of the discriminant, which is needed to find the roots of the equation via the quadratic formula, cannot be evaluated in the realm of real numbers, as the square root of a negative number does not exist among them.

To address this, mathematicians have introduced the imaginary unit \(i\), defined as \(i = \sqrt{-1}\). Hence, if an equation's discriminant is negative, it indicates two complex roots, which will both involve the imaginary unit. These roots will be conjugate pairs, meaning they will have the same real part but opposite imaginary parts. For example, if one root is \(3 + 4i\), the other root will be \(3 - 4i\).

Quadratic Formula

The quadratic formula is a fundamental tool in solving quadratic equations and is derived from the process of completing the square. The formula provides the solutions to any quadratic equation and is expressed as:

\(x = \frac{{-b \pm \sqrt{{b^{2} - 4ac}}}}{{2a}}\)

This formula encapsulates the discriminant within its radical sign: \(b^{2} - 4ac\). Here, \(\pm\) signifies that there will be two possible values for x, which correspond to the two possible roots of the equation. The term \(\sqrt{{b^{2} - 4ac}}\) calculates the discriminant's square root, influencing the nature of the roots:

• If the discriminant is positive, the square root will be a real number, leading to two real solutions.
• If the discriminant is zero, the square root will be zero, resulting in one real solution.
• If the discriminant is negative, the square root will be an imaginary number, yielding two complex solutions.

The simplicity and universality of the quadratic formula make it an essential concept for students to understand.