Problem 77 - A Graphical Approach to Precalculus with Limits

Question

For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverses $$4,5$$

Step-by-Step Solution

Answer

\(a = 1\), \(b = -9\), \(c = 20\).

Step 1: Identify Roots and Use Reverse Zero-Product Property

The roots of the quadratic equation are 4 and 5. According to the zero-product property, the equation can be expressed as \((x - 4)(x - 5) = 0\).

Step 2: Expand the Factored Form

Expand the product \((x - 4)(x - 5)\) to form the quadratic equation. Multiply: \((x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20\).

Step 3: Identify Coefficients

From the expanded form \(x^2 - 9x + 20\), we can identify the coefficients: \(a = 1\), \(b = -9\), \(c = 20\).

Key Concepts

Zero-Product Property

The Zero-Product Property is an essential concept in algebra. It tells us that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is represented as: if \(ab = 0\), then \(a = 0\) or \(b = 0\).

When dealing with quadratic equations, this property helps in finding the solutions or "roots" of the equation. For instance, if we factorize a quadratic equation into two binomials like \((x - a)(x - b) = 0\), it means \(x\) must be equal to either \(a\) or \(b\) to satisfy the equation.

By applying the Zero-Product Property in reverse, as in the given problem, we know that if a quadratic equation has solutions, or roots, we can express it as a product of two binomials. This is an effective way to reconstruct the original equation once the roots are known.

Roots of Equations

Roots of an equation are the values that satisfy the equation when substituted for the variable. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the roots are the values of \(x\) that make the equation true.

For a standard quadratic equation, you may find roots using several methods:

• Factoring the quadratic.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square.

In the given exercise, we're provided with the roots, which are 4 and 5. Knowing these roots helps us set up the equation in its factored form \((x - 4)(x - 5) = 0\), thus making it easier to expand back into standard quadratic form.

Finding the roots enables us to understand important properties of quadratic graphs, such as where they intersect the x-axis.

Factoring

Factoring is a mathematical process used to express an equation as a product of its factors. For quadratic equations, factoring can simplify finding the roots.

Given a quadratic equation in standard form, \(ax^2 + bx + c\), the goal of factoring is to write it in the form \((x - p)(x - q)\) where \(p\) and \(q\) are the roots.

In the exercise, the equation \((x - 4)(x - 5) = 0\) has been expanded to form \(x^2 - 9x + 20\) through multiplication:

• First, distribute: \(x \times x\), \(x \times -5\), \(-4 \times x\), and \(-4 \times -5\).
• Second, combine like terms to get the quadratic \(x^2 - 9x + 20\).

Factoring helps in breaking down more complex expressions and is especially useful when solving equations or simplifying algebraic expressions.

Coefficients

In a quadratic equation expressed as \(ax^2 + bx + c = 0\), the terms \(a\), \(b\), and \(c\) are known as coefficients. These coefficients determine the shape and position of the quadratic curve when plotted on a graph.

- The coefficient \(a\) is the leading coefficient.- The coefficient \(b\) is the linear coefficient.- The coefficient \(c\) is the constant term.

In the example from the exercise, the expanded equation \(x^2 - 9x + 20\) shows that:

• \(a = 1\): indicating a standard "u" shaped parabola since it is positive,.
• \(b = -9\): affecting the horizontal positioning and symmetry of the graph.
• \(c = 20\): the y-intercept where the graph crosses the y-axis.

Understanding coefficients is crucial as they influence the parabola's direction (upward or downward), width, and the points where it crosses the axes.