Problem 47 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$2 x^{2}+x+2=0$$

Step-by-Step Solution

Answer

The quadratic equation \(2x^2 + x + 2 = 0\) has no real solutions, only complex ones, as evidenced by the result of \(-1 \pm \sqrt {-15} / 4\).

Step 1: Identify the coefficients

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

Step 2: Substitute the coefficients into the quadratic formula

Substitute the values of \(a\), \(b\), and \(c\) we identified in step 1 into the quadratic formula, \(-b \pm \sqrt {b^2 - 4ac} / 2a\), to get: \(-1 \pm \sqrt {1^2 - 4*2*2} / 2*2\).

Step 3: Evaluate the quadratic formula

Evaluate the expression in the square root. If this value is negative, then there will be no real solutions, only complex. In this case, the expression evaluates to \(-1 \pm \sqrt {1 - 16} / 4\), which simplifies to \(-1 \pm \sqrt {-15} / 4\), confirming that there are no real solutions. The square root of a negative number does not exist in the real number system, only in the complex number system.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to solve quadratic equations. A quadratic equation is a type of polynomial that can be expressed in the format \( ax^2 + bx + c = 0 \). This formula is particularly useful because it provides a solution for any quadratic equation, regardless of the specific values of \( a \), \( b \), and \( c \). The general form of the quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• The symbol \(\pm\) indicates that there are generally two solutions: one involving addition and the other subtraction.

To use the quadratic formula, substitute the coefficients of your specific quadratic equation into the formula. Then, perform the arithmetic operations to find the solutions. Sometimes, these solutions can include real numbers, and other times they might involve complex numbers.

Real Solutions

Real solutions are the possible x-values for which a quadratic equation equals zero, as long as they exist in the real number system. An important part of finding real solutions using the quadratic formula is the discriminant, found under the square root sign:\[b^2 - 4ac\]

• If the discriminant is greater than zero, there are two distinct real solutions.
• If it is exactly zero, there is one real solution which is a repeated root (also called a double root).
• If the discriminant is less than zero, which happened in the example problem, there are no real solutions.

When the discriminant is less than zero, as \(1 - 16 = -15\) in this quadratic, it means that the equation cannot be solved within the real numbers, leading us to the concept of complex numbers.

Complex Numbers

Complex numbers come into play when the solutions to a quadratic equation are not real. This occurs when the discriminant \(b^2 - 4ac\) is negative. A complex number includes a real part and an imaginary part. The imaginary part is denoted by \(i\), where \(i\) stands for the square root of \(-1\). For instance, if the discriminant is \(-15\), then:\\[\sqrt{-15} = \sqrt{15}i\]In this context, each solution derived from the quadratic formula can have a complex component. The solutions could appear as \(-\frac{1}{4} + \frac{\sqrt{15}i}{4}\) and \(-\frac{1}{4} - \frac{\sqrt{15}i}{4}\). These solutions indicate that there are no x-values on the real number line that make the original quadratic equation equal to zero. Understanding complex numbers extends the set of solutions possible from quadratic equations, allowing us to solve even when the real number system has no solutions.