Problem 67 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}+2.20 x+1.21=0 $$

Step-by-Step Solution

Answer

The equation has one real solution.

Step 1: Identify the Coefficients

The given quadratic equation is \(x^2 + 2.20x + 1.21 = 0\). Identify the coefficients: \(a = 1\), \(b = 2.20\), and \(c = 1.21\).

Step 2: Calculate the Discriminant

Use the formula for the discriminant, \(D = b^2 - 4ac\). Substitute the coefficients: \(D = (2.20)^2 - 4 \cdot 1 \cdot 1.21\).

Step 3: Simplify the Discriminant Expression

Calculate \((2.20)^2 = 4.84\) and \(4 \cdot 1 \cdot 1.21 = 4.84\). Thus, the discriminant \(D = 4.84 - 4.84 = 0\).

Step 4: Interpret the Discriminant

Since the discriminant \(D\) is 0, the quadratic equation has exactly one real solution.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is any equation that can be rearranged in the form of \( ax^2 + bx + c = 0 \). The highest power of the variable \( x \) in a quadratic equation is 2, which is what makes it quadratic. Quadratic equations can often be seen in problems relating to parabolas in geometry or in various calculations involving areas and projectile motion.

These equations always have three components:

• The quadratic coefficient \( a \)
• The linear coefficient \( b \)
• The constant term \( c \)

Each of these coefficients plays a crucial role in determining the properties of the quadratic equation, such as the position and shape of the parabola it represents. Recognizing and understanding these components is the first step in solving or analyzing a quadratic equation.

Coefficients Identification Made Simple

In the quadratic equation \( x^2 + 2.20x + 1.21 = 0 \), identifying coefficients is straightforward. The coefficient of \( x^2 \) is \( a \) and here it equals 1. The coefficient of \( x \) is \( b \), which is 2.20. The constant term is \( c \), valued at 1.21.

Let's break down how these coefficients are identified:

• Look for the term multiplied by \( x^2 \). That's your \( a \).
• Find the term multiplied by \( x \), and you have your \( b \).
• The standalone number without \( x \) is your \( c \).

Knowing these coefficients is needed to perform any further analysis, such as calculating the discriminant, which gives insights into the nature of the solutions of the quadratic equation.

The Role of Real Solutions

Real solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These solutions can be found using various methods like factoring, completing the square, or applying the quadratic formula.

The work in determining whether a quadratic equation has real solutions often starts with analyzing the discriminant. The discriminant helps us predict the nature of the solutions without actually finding them precisely.

When a quadratic equation has real solutions, it means the graph of the equation (a parabola) will intersect the x-axis. The number of intersections can be one, two, or none, based on the value of the discriminant.

Interpreting the Discriminant

The discriminant, denoted by \( D \), is used to determine the nature of the roots of a quadratic equation without solving it completely. It is calculated using the formula \( D = b^2 - 4ac \).

Once calculated, the discriminant can be interpreted as follows:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, meaning it touches the x-axis at one point.
• If \( D < 0 \), the equation has no real roots, indicating the parabola does not intersect the x-axis.

In our given equation \( x^2 + 2.20x + 1.21 = 0 \), we found \( D = 0 \), thus implying that it has precisely one real solution. Understanding the discriminant allows students to grasp the potential outcomes of solving quadratic equations and predict the nature of the solutions confidently.