Problem 12 - Algebra 2

Question

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(4 x^{2}+4 x+1=0\)

Step-by-Step Solution

Answer

The discriminant is 0; there is one real double root.

Step 1: Identify the coefficients

The given quadratic equation is \( 4x^2 + 4x + 1 = 0 \). The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. Here, \( a = 4 \), \( b = 4 \), and \( c = 1 \).

Step 2: Calculate the discriminant

The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \). Substituting the values, we get \( \Delta = 4^2 - 4 \times 4 \times 1 = 16 - 16 = 0 \).

Step 3: Analyze the discriminant

The value of the discriminant is 0. When the discriminant is 0, the quadratic equation has exactly one real root, and it is a double root or repeated root.

Key Concepts

Understanding the Discriminant

The discriminant is a fundamental component when working with quadratic equations. It is found using the formula \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

Calculating the discriminant helps in predicting the nature of the roots of the equation without actually solving it. There are three possible scenarios for the value of the discriminant:

• If \( \Delta > 0 \): The quadratic equation has two distinct real roots.
• If \( \Delta = 0 \): The equation has exactly one real root, which is also a double root - meaning the graph of the equation touches the x-axis at one point.
• If \( \Delta < 0 \): The equation has no real roots but rather two complex roots, which means the graph does not intersect the x-axis.

Calculating the discriminant and analyzing its value can quickly provide insights into the solution set of a quadratic equation.

Roots of Quadratic Equations

Roots of quadratic equations are the solutions or the x-values that make the equation equal zero. These can be found using various methods, such as factoring, completing the square, or using the quadratic formula.

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a versatile tool that directly utilizes the discriminant. Depending on the sign and value of the discriminant, you can determine the type of roots:

• When \( \Delta > 0 \), two distinct real roots exist. The roots are found by solving with both the plus and minus in the formula.
• When \( \Delta = 0 \), there is one real and repeated root. This is because both solutions of the quadratic formula will be the same.
• When \( \Delta < 0 \), the roots are complex, coming in the form of a complex conjugate pair.

Understanding the roots is crucial as they represent the points where the graph of the quadratic equation intersects or touches the x-axis.

Coefficient Identification in Quadratic Equations

Identifying coefficients in a quadratic equation is the first and indispensable step in analyzing the equation. In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the coefficients are:

• \( a \): the coefficient of \( x^2 \) which determines the opening direction and width of the parabola.
• \( b \): the coefficient of \( x \) which influences the direction and vertex of the parabola.
• \( c \): the constant term which affects the y-intercept of the graph.

Accurate identification of these coefficients allows you to substitute the correct values into the discriminant formula and the quadratic formula.

Therefore, correctly picking \( a \), \( b \), and \( c \) is essential to solving and understanding quadratic equations. It paves the way for deeper analysis and predictions about the characteristics of the quadratic's graph.