Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{3} x^{2}-\frac{1}{3} x=24$$

Step-by-Step Solution

Answer

The real solutions are \(x = 9\) and \(x = -8\).

Step 1: Clear Fractions

To eliminate the fractions, multiply both sides of the equation by 3. This gives:\[3 \left( \frac{1}{3} x^2 - \frac{1}{3} x \right) = 3 \times 24\]Simplifying, we have:\[x^2 - x = 72\]

Step 2: Bring to Standard Form

Rearrange the equation to the standard quadratic form by moving all terms to one side:\[x^2 - x - 72 = 0\]

Step 3: Factor the Quadratic Equation

To factor \(x^2 - x - 72 = 0\), we need two numbers that multiply to -72 and add to -1. These numbers are 8 and -9. Therefore, the equation can be factored as:\[(x - 9)(x + 8) = 0\]

Step 4: Solve for x

Set each factor equal to zero to solve for \(x\):1. \(x - 9 = 0\) \implies \(x = 9\)2. \(x + 8 = 0\) \implies \(x = -8\)The solutions are \(x = 9\) and \(x = -8\).

Step 5: Graphical Representation

The quadratic equation \(x^2 - x - 72 = 0\) is a parabola that opens upwards because the coefficient of \(x^2\) is positive. The vertex is at \(x = \frac{1}{2}\) (calculated as \(\frac{-b}{2a}\) with \(a=1, b=-1\)). The x-intercepts, which are the real solutions, are at \(x = 9\) and \(x = -8\), confirming our solutions.

Key Concepts

Factoring

Factoring quadratic equations is a crucial method for solving them. When we have an equation like the standard quadratic form \(x^2 - x - 72 = 0\), our task is to express it as a product of two binomials. This is based on finding two numbers that multiply to the constant term, here \(-72\), and add to the linear coefficient, here \(-1\). In factoring, you will often:

• Look for two numbers that multiply to the constant term, in this case \(-72\).
• Ensure these numbers also add up to the linear coefficient, \(-1\).

The numbers found are \(8\) and \(-9\), since \(8 \times -9 = -72\) and \(8 + (-9) = -1\). Thus, we factor the equation into \((x - 9)(x + 8) = 0\). Factoring simplifies solving the equation, as each binomial set to zero reveals the potential solutions for \(x\).

Graphical Representation

Understanding the graphical representation of a quadratic equation can enhance our grasp of its solutions. Upon conversion into standard form, our equation becomes \(x^2 - x - 72 = 0\), plotted as a parabola. This parabola opens upwards because the coefficient of \(x^2\) is positive.When graphing:

• The parabola’s shape depends on the coefficient of \(x^2\). If it's positive, it opens upward; if negative, downward.
• Identify the vertex, calculated as \(x = \frac{-b}{2a}\) for the quadratic \(ax^2 + bx + c\). Here, the vertex occurs at \(x = \frac{1}{2}\).

The real solutions \(x = 9\) and \(x = -8\) will appear as intercepts where the parabola crosses the x-axis. This visual check supports our algebraic solutions, providing both an analytical and graphical means to confirm these solutions.

Solutions of Quadratics

Solving quadratic equations offers us insight into the potential values of \(x\) that satisfy the equation. When the quadratic is factored, like \((x - 9)(x + 8) = 0\), each factor is set to zero to find these solutions.This process involves:

• Setting each binomial factor equal to zero: \(x - 9 = 0\) yields \(x = 9\), and \(x + 8 = 0\) results in \(x = -8\).
• These solutions are immediately applicable because substituting them into the original equation satisfies it completely.

By solving the factors for zero, we make the equation manageable and verify each solution. Always ensure that these solutions make sense in the context of the original equation, solidifying their validity. These values can also be double-checked against the graphical representation, as solutions correspond to x-intercepts of the graphed parabola.