Problem 70 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the discriminant to determine the number of real solutions of the equation. $$ (2 p+1)^{2}-3(2 p+1)+4=0 $$

Step-by-Step Solution

Answer

The given equation is \((2p+1)^2 - 3(2p+1) + 4 = 0\). Upon expanding and combining terms, we get the quadratic equation \(4p^2 - 2p + 2 = 0\), with \(a = 4\), \(b = -2\), and \(c = 2\). The discriminant (\(D\)) is given by the formula \(D = b^2 - 4ac = (-2)^2 - 4(4)(2) = 4 - 32 = -28\). Since the discriminant is negative (\(D < 0\)), the equation has no real solutions (complex roots).

Step 1: Expand the Equation

Begin by expanding the equation \((2p+1)^2 - 3(2p+1) + 4 = 0\): $$ (2p+1)^2 - 3(2p+1) + 4 = (4p^2 + 4p + 1) - (6p +3) + 4 $$

Step 2: Collect Terms

Combine like terms to obtain the quadratic equation: $$ 4p^2 + 4p + 1 - 6p - 3 + 4 = 4p^2 - 2p + 2 = 0 $$ The values of \(a\), \(b\), and \(c\) are 4, -2, and 2, respectively.

Step 3: Find the Discriminant

Calculate the discriminant using the formula \(D = b^2 - 4ac\), where \(a = 4\), \(b = -2\), and \(c = 2\): $$ D = (-2)^2 - 4(4)(2) = 4 - 32 = -28 $$

Step 4: Determine the Number of Real Solutions

Since the discriminant is negative (i.e., \(D < 0\)), the quadratic equation has no real solutions (complex roots).

Key Concepts

Real Solutions of Quadratic Equations

Understanding when a quadratic equation has real solutions is a fundamental aspect of algebra. A quadratic equation is typically written in the form
\( ax^2 + bx + c = 0 \), where
\( a \), \( b \), and \( c \), are constants with
\( a \eq 0 \).

The key to determining the nature of the roots—whether they are real or complex—is the discriminant, denoted by \( D \). The discriminant is calculated using the formula \( D = b^2 - 4ac \). For real solutions to exist, the discriminant must be greater than or equal to zero. Here are the possibilities:

• \(\textbf{If } D > 0:\) The equation has two distinct real solutions.
• \(\textbf{If } D = 0:\) There is exactly one real solution (also known as a repeated or double root).
• \(\textbf{If } D < 0:\) The equation has no real solutions; instead, there are two complex solutions.

By calculating the discriminant, we can predict the nature of the solutions without actually solving the equation.

Expanding Quadratic Equations

To solve a quadratic equation, you may need to expand the expressions to bring them into a standard quadratic form. Expanding involves simplifying mathematical expressions and removing brackets.

Let's consider \( (ax+b)^2 \). It expands to \( a^2x^2 + 2abx + b^2 \), following the principles of algebraic multiplication and addition. Similarly, if we have a quadratic equation with nested expressions like \( (2p+1)^2 - 3(2p+1) + 4 = 0 \) from the exercise, we must expand these expressions to find the coefficients of the standard form.

For the given equation:

• First, you expand the squared term, \( (2p+1)^2 \) using the formula \( (a+b)^2 = a^2 + 2ab + b^2 \).
• Then, distribute the \( -3 \) across the expression \( (2p+1) \).
• Finally, combine all like terms, which will give us a clean quadratic equation in the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are easily identifiable.

This step is crucial before calculating the discriminant, as it forms the basis for identifying the coefficients used within that calculation.

Calculating the Discriminant

Once you have a quadratic equation in standard form, calculating the discriminant is straightforward but crucial to identify the types of solutions the equation has. As mentioned earlier, the discriminant is found using the formula \( b^2 - 4ac \).

Here's how it works:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c \).
• Substitute these values into the discriminant formula.
• Calculate the value of the discriminant to use it as an indicator of the nature of the roots.

In our exercise, after expanding and simplifying the original expression, we identified \( a = 4 \), \( b = -2 \), and \( c = 2 \). Substituting them gives us \( D = (-2)^2 - 4(4)(2) \), which simplifies to \( D = 4 - 32 = -28 \). The negative value of the discriminant points to the absence of real solutions and suggests the presence of complex roots instead.

Thus, calculating the discriminant serves as a powerful tool in assessing the solutions to a quadratic equation, helping us understand the equation's properties without the need for finding the roots explicitly.