Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$11 x^{2}-3 x+2=4 x+1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{7 + \sqrt{5}}{22}\) and \(x = \frac{7 - \sqrt{5}}{22}\).

Step 1: Rewrite the Equation

Start by rearranging the given equation so that one side is zero. The original equation is \(11x^2 - 3x + 2 = 4x + 1\). Subtract \(4x + 1\) from both sides to get: \[11x^2 - 3x + 2 - 4x - 1 = 0\] which simplifies to \[11x^2 - 7x + 1 = 0\].

Step 2: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For the equation \(11x^2 - 7x + 1 = 0\), identify \(a = 11\), \(b = -7\), and \(c = 1\). Substitute these values into the formula: \[x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 11 \cdot 1}}{2 \cdot 11}\].

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\), which will help us identify the nature of the roots: \((-7)^2 - 4 \times 11 \times 1 = 49 - 44 = 5\). As the discriminant is positive, the equation has two distinct real solutions.

Step 4: Solve for the Roots

Now substitute the discriminant back into the quadratic formula: \[x = \frac{7 \pm \sqrt{5}}{22}\]. This gives us two solutions: \[x = \frac{7 + \sqrt{5}}{22}\] and \[x = \frac{7 - \sqrt{5}}{22}\].

Step 5: Graphical Representation

To graphically represent the solutions, draw the parabola \(y = 11x^2 - 7x + 1\) and the line \(y = 0\). The x-coordinates where the parabola intersects the x-axis represent the solutions: \(x = \frac{7 + \sqrt{5}}{22}\) and \(x = \frac{7 - \sqrt{5}}{22}\). Use graphing software or a calculator to aid with the graph.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \(ax^{2} + bx + c = 0\). It provides a straightforward method to find the values of \(x\) that satisfy this equation. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula saves time and reduces error, especially with complex fractions and roots.
Remembering each component:

• \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^2 + bx + c\)
• \(-b\) indicates the change in sign of the coefficient \(b\)
• \(\pm\) denotes that there are usually two solutions: one by adding, the other by subtracting

Using this approach to solve the exercise’s equation, we substitute \(a = 11\), \(b = -7\), and \(c = 1\) into the formula. This reveals the presence of two potential solutions for \(x\).

Discriminant

The discriminant in the quadratic formula is the part under the square root, \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the quadratic equation.
The value of the discriminant can tell us the following:

• If it is positive, the equation has two distinct real solutions.
• If it is zero, the quadratic has exactly one real solution, or a repeated root.
• If negative, there are no real solutions, only complex or imaginary solutions.

In the exercise, the discriminant is calculated as \((-7)^2 - 4 \times 11 \times 1 = 5\). This positive value suggests that the quadratic equation has two separate and real solutions. This information is crucial before actual computations to determine the expected types of solutions.

Graphical Solution

Graphical solutions provide a visual interpretation of where the solutions to a quadratic equation lie. This involves plotting the quadratic function as a parabola on the graph, such as \(y = 11x^2 - 7x + 1\) from the exercise.
Follow these steps for a graphical solution:

• Set up your graph with appropriate scales for \(x\) and \(y\).
• Graph the quadratic equation, recognizing that the parabola opens upwards as the coefficient of \(x^2\) (\(11\)) is positive.
• Identify where the parabola intersects the \(x\)-axis. These intersection points represent where \(y = 0\), or the solutions of the quadratic equation.

In the exercise, plotting helps confirm the analytical solutions \(x = \frac{7 + \sqrt{5}}{22}\) and \(x = \frac{7 - \sqrt{5}}{22}\). Seeing these intersection points makes the otherwise abstract solutions tangible.