Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-3)=2$$

Step-by-Step Solution

Answer

Solutions: \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).

Step 1: Expand the Equation

Starting with the equation given, expand the left-hand side: \(x(x-3) = x^2 - 3x\). So, the equation becomes \(x^2 - 3x = 2\).

Step 2: Move all terms to one side

Subtract 2 from both sides of the equation to set it to zero: \(x^2 - 3x - 2 = 0\). Now the equation is in a standard quadratic form.

Step 3: Apply the Quadratic Formula

The standard form is \(ax^2 + bx + c = 0\), so \(a = 1\), \(b = -3\), \(c = -2\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Calculate the Discriminant

Find the discriminant \(b^2 - 4ac\): \((-3)^2 - 4 \cdot 1 \cdot (-2) = 9 + 8 = 17\). Since the discriminant is positive, there are two real solutions.

Step 5: Solve for x using the Quadratic Formula

Substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{17}}{2 \cdot 1} = \frac{3 \pm \sqrt{17}}{2} \]. So, the solutions are \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).

Step 6: Graphical Representation

Graph the function \(f(x) = x^2 - 3x - 2\) and find the x-intercepts. These intercepts occur where \(f(x) = 0\), confirming the solutions \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
To find the roots (solutions) of the equation, you can use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works step-by-step:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Plug these values into the quadratic formula.
• Solve for \(x\) to find the possible solutions.

Understanding this formula can greatly simplify the process of solving quadratic equations, especially when factoring is challenging.

Discriminant

The discriminant is a crucial part of the quadratic formula, found under the square root sign. It is given by
\(b^2 - 4ac\).
The discriminant helps in determining the nature of the solutions of a quadratic equation.

• If the discriminant is positive (>0), there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (the roots are the same).
• If the discriminant is negative (<0), there are no real solutions (the solutions are complex numbers).

In our example, the discriminant is 17, which is positive, indicating two real solutions.

Real Solutions

Real solutions are the x-values where a quadratic equation equals zero. They can be visualized as the points where the graph of the quadratic function crosses the x-axis on a coordinate plane.
These points are also known as the roots or x-intercepts of the function.

• To find the real solutions, compute them using the quadratic formula.
• If the discriminant is positive, expect two different real numbers as solutions, like in the example, \(x = \frac{3 + \sqrt{17}}{2}\) and \(x = \frac{3 - \sqrt{17}}{2}\).

Real solutions are essential for understanding the behavior of quadratic functions across different intervals of x-values.

Graphical Representation

The graphical representation of a quadratic function helps visualize its solutions.
The graph of a quadratic equation forms a parabola. This parabola can either open upwards or downwards, depending on the sign of the coefficient \(a\).Here is what to look for when graphing:

• Find the x-intercepts of the parabola, as these represent the real solutions.
• The vertex of the parabola gives insight into the maximum or minimum value of the function.
• Notice the axis of symmetry that divides the parabola into mirror images.

In our example, the quadratic function \(f(x) = x^2 - 3x - 2\) has x-intercepts at \(x = \frac{3 + \sqrt{17}}{2}\) and \(x = \frac{3 - \sqrt{17}}{2}\), confirming the solutions through the visual crossing of the x-axis.