Problem 64 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(x^{2}+2.21 x+1.21=0\)

Step-by-Step Solution

Answer

The equation has 2 distinct real solutions.

Step 1: Identify the coefficients

The given quadratic equation is \(x^{2} + 2.21x + 1.21 = 0\). Here, \(a = 1\), \(b = 2.21\), and \(c = 1.21\).

Step 2: Write the discriminant formula

The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\).

Step 3: Substitute values into the discriminant formula

Substitute \(b = 2.21\), \(a = 1\), and \(c = 1.21\) into the discriminant formula: \(D = (2.21)^2 - 4(1)(1.21)\).

Step 4: Calculate the discriminant

First, calculate \((2.21)^2 = 4.8841\). Then calculate \(4 \times 1 \times 1.21 = 4.84\). Therefore, the discriminant \(D = 4.8841 - 4.84 = 0.0441\).

Step 5: Determine the number of real solutions

Since the discriminant \(D = 0.0441\) is greater than zero, the quadratic equation has 2 distinct real solutions.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in algebra, defining any polynomial equation of the form \( ax^2 + bx + c = 0 \). These equations represent parabolic curves when plotted on a graph. The 'quadratic' term arises because the highest degree of the variable, usually \( x \), is squared. This makes them slightly more complex than linear equations, but they can always be solved using consistent methods. Key components of a quadratic equation are:

• \( a \), the quadratic coefficient, which cannot be zero.
• \( b \), the linear coefficient.
• \( c \), the constant term.

Each of these coefficients plays a crucial role in defining the curve's shape and position on a graph. The solution to a quadratic can take the form of real or complex numbers, depending on the discriminant value.

Number of Real Solutions in Quadratic Equations

The number of real solutions a quadratic equation possesses depends on the value of its discriminant. The discriminant, denoted as \( D \), is calculated using the formula \( D = b^2 - 4ac \). This specific value tells you whether the parabola will intersect the \( x \)-axis and how many times.Here are the scenarios:

• If \( D > 0 \), there are 2 distinct real solutions. The parabola crosses the \( x \)-axis at two points.
• If \( D = 0 \), there is exactly 1 real solution, also called a repeated or double root. The parabola just touches the \( x \)-axis.
• If \( D < 0 \), there are no real solutions. The parabola does not cross the \( x \)-axis, and the solutions are complex numbers.

These outcomes provide a quick way to classify quadratic equations based on their graphical representation and solution type.

Calculating Discriminants

Calculating the discriminant of a quadratic equation is a straightforward process if done step by step. For the equation example \( x^2 + 2.21x + 1.21 = 0 \), you need to:1. Identify the coefficients: \( a = 1 \), \( b = 2.21 \), and \( c = 1.21 \). 2. Apply these values to the discriminant formula: \( D = b^2 - 4ac \).3. Substitute the coefficients into the formula: \( D = (2.21)^2 - 4 \times 1 \times 1.21 \).4. Perform the calculations:

• First, calculate \( (2.21)^2 = 4.8841 \).
• Next, calculate \( 4 \times 1.21 = 4.84 \).
• Subtract these results to get \( D = 4.8841 - 4.84 = 0.0441 \).

The final discriminant, 0.0441, is greater than zero, confirming the equation has two distinct real solutions. This simple calculation allows you to determine the nature of the solutions without fully solving the equation.