Problem 14 - Algebra 1

Question

Using THE DISCRIMINANT Tell if the equation has two solutions, one solution, or no real solution. $$3 x^{2}-6 x+3=0$$

Step-by-Step Solution

Answer

The given quadratic equation \(3x^2 - 6x + 3 = 0\) has one real solution, because the discriminant equals 0.

Step 1: Identify coefficients

Identify the coefficients in the quadratic equation, where \(a = 3, b = -6, c = 3\).

Step 2: Calculate the discriminant

Calculate the discriminant using the formula \(b^2 - 4ac = (-6)^2 - 4*3*3\).

Step 3: Analyze Discriminant

Analyze the calculated value of the discriminant to determine the number of roots. If it's greater than 0, there are two solutions. If equal to 0, there is one solution, and if less than 0, there is no real solution

Key Concepts

Quadratic Equations

Understanding quadratic equations is fundamental in algebra. These equations are in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are numerical coefficients and \( a \) is not zero. The reason \( a \) cannot be zero is that this would eliminate the \( x^2 \) term, leaving a linear equation.

A quadratic equation can represent a parabola when graphed on a coordinate plane. The direction (upward or downward facing) and the width of the parabola vary depending on the coefficients. To solve a quadratic equation, you can use several methods such as factoring, completing the square, using the quadratic formula, or graphing. Each method serves different scenarios and ease of use, but in more complex situations, the quadratic formula is often the most reliable method.

Factoring and Formula Methods

For example, when the equation can easily be factored into binomials, factoring is the simplest method. However, if factoring is cumbersome or impossible, the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \) becomes the best course of action.
To use the quadratic formula, we must calculate the discriminant \( \Delta = b^2 - 4ac \), which determines the nature of the roots of the quadratic equation.

Discriminant Analysis

Discriminant analysis revolves around the discriminant of a quadratic equation, symbolized as \( \Delta \). It is given by the formula \( \Delta = b^2 - 4ac \), where \( b \) and \( c \) are the coefficients of \( x \) and the constant term, respectively, in the quadratic equation \( ax^2 + bx + c = 0 \).

The discriminant plays a pivotal role in predicting the nature and number of roots that a quadratic equation will have without necessarily computing them. The value of the discriminant reveals:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root (also known as a repeated or double root).
• If \( \Delta < 0 \), the equation has no real roots; instead, it has two complex roots.

In our example \( 3x^2 - 6x + 3 = 0 \), after identifying the coefficients \( a = 3 \), \( b = -6 \), and \( c = 3 \), we calculate the discriminant as \( (-6)^2 - 4 \cdot 3 \cdot 3 = 0 \). This indicates the parabola described by the function touches the x-axis at exactly one point, resulting in one real root. Factors such as these are crucial when solving real-world problems where the distinction between different types of solutions is necessary.

Real Solutions of Quadratics

Considering the importance of real solutions, they are solutions to quadratic equations where the answers are real numbers, as opposed to complex numbers that have both real and imaginary parts. Real solutions occur when the graph of the quadratic equation, the parabola, intersects the x-axis. The nature of these intersections and the corresponding solutions are determined by the discriminant of the equation.

To elucidate, when the discriminant \( \Delta = 0 \), the parabola exactly touches the x-axis at a single point known as the vertex. Therefore, there is one real solution, which is also the axis of symmetry of the parabola. Conversely, when \( \Delta > 0 \), the parabola intersects the x-axis at two distinct points, thus yielding two real solutions.

Solving for Real Roots

In the case where the discriminant \( \Delta < 0 \), since the parabola does not intersect the x-axis at all, there are no real solutions, and the equation has complex roots that are not graphed on a standard x-y coordinate grid. Students should be comfortable with the concept that not all quadratic equations will produce real-number solutions, and this understanding is fundamental when interpreting graphs or formulating equations to fit real-world scenarios.