Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+5)(x-6)=(2 x-1)(x-4)$$

Step-by-Step Solution

Answer

There are no real solutions, as the discriminant is negative.

Step 1: Expand the Equations

First, expand both sides of the equation. The left-hand side \((x+5)(x-6)\) becomes \(x^2 - 6x + 5x - 30\), which simplifies to \(x^2 - x - 30\). The right-hand side, \((2x-1)(x-4)\), expands to \(2x^2 - 8x - x + 4\), simplifying to \(2x^2 - 9x + 4\).

Step 2: Bring to One Side

Subtract the right-hand side equal from the left:\((x^2 - x - 30) - (2x^2 - 9x + 4) = 0\).This results in:\(-x^2 + 8x - 34 = 0\).

Step 3: Simplify the Equation

To make the equation easier to manage, multiply through by \(-1\) so that the leading coefficient is positive: \(x^2 - 8x + 34 = 0\).

Step 4: Calculate the Discriminant

Determine if real solutions exist by calculating the discriminant of the quadratic equation. The discriminant is given by \(b^2 - 4ac\), where \(a = 1\), \(b = -8\), and \(c = 34\).So, \((-8)^2 - 4(1)(34) = 64 - 136 = -72\).

Step 5: Interpret the Discriminant

Since the discriminant \(-72\) is less than zero, the quadratic equation \(x^2 - 8x + 34 = 0\) has no real solutions. This means the graph of the quadratic equation will not intersect the x-axis.

Key Concepts

Discriminant

The discriminant is a crucial part of understanding quadratic equations. It helps us determine the nature and number of solutions. For a quadratic equation in the form of \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated as \(b^2 - 4ac\).

• If \(D > 0\), the quadratic equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution; the vertex of the parabola touches the x-axis.
• If \(D < 0\), there are no real solutions, meaning the graph of the equation does not intersect the x-axis at all.

In our problem, the discriminant was calculated as \(-72\). Since this value is less than zero, the quadratic equation \(x^2 - 8x + 34 = 0\) lacks real solutions, indicating a graph that never touches the x-axis.

Graphical Solutions

A graphical solution gives a visual representation of the solutions to an equation. For quadratic equations, the graph is typically a parabola.

• A parabola that crosses the x-axis at two points corresponds to a quadratic equation with two real solutions.
• If it touches the x-axis at one point, there is one real solution.
• If it does not interact with the x-axis, there are no real solutions.

Since our discriminant \(-72\) indicates there are no real solutions, graphing the equation \(x^2 - 8x + 34 = 0\) would show a parabola entirely above or below the x-axis without intersecting it.
This highlights how powerful and intuitive graphical solutions can be in understanding quadratic equations.

Real Solutions

Real solutions are the x-values for which the quadratic equation equals zero. These are the points where the graph intersects the x-axis. Real solutions can be determined from the discriminant and graphically confirmed.
When the discriminant is positive, two real solutions exist. If it's zero, the quadratic equation has exactly one real solution. For negative discriminants, like the one in our problem, no real solutions exist.
Real solutions are essential in real-world applications, such as physics or engineering, where they can represent time, distance, or other measurable quantities. Unfortunately, when a quadratic has no real solutions, it signifies no real-world scenario matches this equation without further context or constraints.

Polynomial Expansion

Polynomial expansion involves expressing a product of sums as a sum of terms. In our equation, we started with \((x+5)(x-6)\) on one side and \((2x-1)(x-4)\) on the other.
To expand, you multiply each term in the first polynomial by each term in the second. This process gives you new terms, which you then simplify:

• From \((x+5)(x-6)\), the expansion is \(x^2 - 6x + 5x - 30\), which simplifies to \(x^2 - x - 30\).
• From \((2x-1)(x-4)\), you get \(2x^2 - 8x - x + 4\), simplifying to \(2x^2 - 9x + 4\).

Expanding polynomials is crucial for simplifying and solving equations, converting complex expressions into a manageable form, and deriving solutions like our eventual quadratic form \(x^2 - 8x + 34 = 0\). This step is foundational in algebra and pivotal for progressing with mathematical problem-solving.