Problem 2 - College Algebra

Question

When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form \(a x^{2}+b x+c=0\) we may graph the equation \(y=a x^{2}+b x+c\) and have no zeroes \((x\) -intercepts \()\)

Step-by-Step Solution

Answer

A quadratic equation has up to two solutions, but there may be none if the discriminant is negative.

Step 1: Identify the Type of Equation

We are dealing with a quadratic equation in the standard form \(ax^2 + bx + c = 0\). A quadratic equation usually has two potential solutions or roots because it's a second-degree polynomial.

Step 2: Solving the Quadratic Equation

The possible solutions or "roots" of a quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The term under the square root, \(b^2 - 4ac\), is called the discriminant.

Step 3: Analyze the Discriminant

The nature and number of solutions depend on the discriminant \(b^2 - 4ac\): - If it is positive, there are two distinct real solutions.- If it is zero, there is exactly one real solution (the parabola touches the x-axis).- If it is negative, there are no real solutions (no x-intercepts).

Step 4: Consider Graphing the Quadratic

When we graph \(y = ax^2 + bx + c\), the solutions to the quadratic equation correspond to the x-intercepts of the graph (where \(y = 0\)). If the discriminant is negative, the parabola does not intersect the x-axis, indicating there are no real zeroes or x-intercepts.

Step 5: Conclusion

We always start out seeking two solutions because a quadratic is a degree 2 polynomial, suggesting up to two roots. However, graphing can reveal that there are sometimes no x-intercepts if the discriminant is negative.

Key Concepts

Standard Form

The standard form of a quadratic equation is written as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, where:

• \( a \) is the coefficient of the squared term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term.

The term 'standard form' is essential because it provides a structured way to recognize and solve quadratic equations. All quadratic equations can be transformed into this format, making them easier to analyze and solve. Understanding this form is the first step in solving quadratic equations and helps in identifying key components such as the coefficients and the constant.

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula directly calculates the roots of a quadratic equation in standard form. Here's how it works:

• Take the parameters (\( a \), \( b \), and \( c \)) from the standard form equation.
• Substitute them into the formula.
• The "\( \pm \)" symbol means you get two solutions: one with addition and one with subtraction.

The quadratic formula is incredibly valuable because it can solve any quadratic equation, regardless of whether the solutions are rational or irrational. It is a universal method that does not require factoring or completing the square, making it especially useful in complex situations.

Discriminant

The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It reveals a lot about the nature of the solutions:

• If the discriminant is positive (\( b^2 - 4ac > 0 \)), there are two distinct real solutions. This means the parabola will intersect the x-axis at two points.
• If the discriminant is zero (\( b^2 - 4ac = 0 \)), there is exactly one real solution. The parabola touches the x-axis at a single point, known as a double root.
• If the discriminant is negative (\( b^2 - 4ac < 0 \)), there are no real solutions. That means the parabola does not touch the x-axis at all.

The discriminant is crucial because it tells us whether we should expect to find real x-intercepts on the graph of the quadratic equation. By evaluating it before solving, you can anticipate the number of solutions you will obtain.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation and are real numbers. In the context of quadratic equations, having no, one, or two real solutions has different meanings:

• Two real solutions: The graph intersects the x-axis at two distinct points. This typically happens when the discriminant is positive.
• One real solution: The graph is tangent to the x-axis, touching it at exactly one point. This occurs when the discriminant is zero.
• No real solutions: The graph does not intersect the x-axis anywhere, indicating the discriminant is negative.

Recognizing real solutions is essential for graphing and real-world problem-solving, as they represent the visible intersections of the parabola with the x-axis on the coordinate plane. Understanding the potential number of real solutions beforehand aids in both algebraic calculations and graphical analysis.