Problem 41 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{1 + \sqrt{5}}{2}\) and \(x = \frac{1 - \sqrt{5}}{2}\).

Step 1: Expand the Equation

The given equation is:\[ x(x - 1) = 1 \]First, expand the left side by distributing the \(x\):\[ x^2 - x = 1 \]

Step 2: Rearrange into Standard Quadratic Form

Next, move all terms to one side of the equation to form a standard quadratic equation:\[ x^2 - x - 1 = 0 \]

Step 3: Use the Quadratic Formula

Identify the coefficients from the quadratic equation \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -1\), and \(c = -1\). Use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the coefficients:\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-1)}}{2 \times 1} \]

Step 4: Calculate the Discriminant and Solutions

Calculate the discriminant:\[ b^2 - 4ac = 1 + 4 = 5 \]Now substitute the discriminant into the quadratic formula:\[ x = \frac{1 \pm \sqrt{5}}{2} \]This results in two solutions:\[ x = \frac{1 + \sqrt{5}}{2} \] and \[ x = \frac{1 - \sqrt{5}}{2} \]

Step 5: Verify Solutions Graphically

To support these solutions graphically, consider the quadratic function \( f(x) = x^2 - x - 1 \). Plot the function and observe where it crosses the x-axis.The graph should intersect the x-axis at the points \( x = \frac{1 + \sqrt{5}}{2} \) and \( x = \frac{1 - \sqrt{5}}{2} \), confirming the real solutions.

Key Concepts

Graphical Solution

To solve a quadratic equation graphically, we translate the equation into a visual graph. This involves plotting the function associated with the quadratic equation, usually in the form of \(f(x) = ax^2 + bx + c\).
For the equation \(x(x - 1) = 1\), after rearranging and simplifying into the standard form, we graph \(f(x) = x^2 - x - 1\).

• The solutions to the equation are where the graph crosses the x-axis. These are the x-intercepts.
• This visual method helps confirm solutions obtained analytically, as seen when the graph of \(f(x)\) intersects the x-axis at points \(x = \frac{1 + \sqrt{5}}{2}\) and \(x = \frac{1 - \sqrt{5}}{2}\).

Observing the graph, you can see these intersections clearly mark the real roots. Graphical solutions provide an intuitive understanding and verification of the solutions.

Quadratic Formula

The quadratic formula is a powerful tool to find roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• This formula calculates all possible solutions, known as roots, for the quadratic equation.

In our case, the equation \(x^2 - x - 1 = 0\) gives us values \(a = 1\), \(b = -1\), and \(c = -1\).
Plugging these into the quadratic formula provides the solutions. Therefore, this method allows solving without needing any graphic, although it is always good to verify it visually.

Discriminant

The discriminant is a key concept in understanding the nature of the roots, calculated from the expression \(b^2 - 4ac\) within the quadratic formula.
It tells us how many real roots the quadratic equation has:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root (the roots are repeated).
• If it is negative, there are no real roots (the roots are complex numbers).

For \(x^2 - x - 1 = 0\), the discriminant is \(5\), positive, indicating two distinct real solutions. This supports the solutions derived earlier using both the quadratic formula and the graphical solution. Understanding the discriminant provides insight beyond just calculating zeros; it also gives a quick sense of what type of solutions to expect.