Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x(12 x+11)=-2$$

Step-by-Step Solution

Answer

The real solutions are \(x = -\frac{1}{3}\) and \(x = -\frac{2}{3}\).

Step 1: Write the Equation in Standard Form

Start by expanding and rearranging the given equation to the standard quadratic form \(ax^2 + bx + c = 0\). The equation given is \(x(12x + 11) = -2\). Expand and rearrange it: \(12x^2 + 11x + 2 = 0\).

Step 2: Identify Coefficients

In the standard form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 12\), \(b = 11\), \(c = 2\).

Step 3: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plug in the values of \(a\), \(b\), and \(c\): \(x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 12 \cdot 2}}{2 \cdot 12}\).

Step 4: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac = 11^2 - 4 \times 12 \times 2 = 121 - 96 = 25\). Since the discriminant is positive, there are two real solutions.

Step 5: Solve for Roots

Calculate the roots using the quadratic formula: \(x = \frac{-11 \pm \sqrt{25}}{24}\). This simplifies to \(x = \frac{-11 + 5}{24}\) and \(x = \frac{-11 - 5}{24}\). Therefore, the solutions are \(x = -\frac{1}{3}\) and \(x = -\frac{2}{3}\).

Step 6: Verify Solutions Graphically

Graph the quadratic function \(y = 12x^2 + 11x + 2\) and confirm it intersects the x-axis at \(x = -\frac{1}{3}\) and \(x = -\frac{2}{3}\), verifying the real solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). This formula allows you to determine the solutions for \( x \) without needing to factorize the quadratic expression. Given a quadratic equation, the solutions can be calculated using the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]Here's how the formula works:

• \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation in standard form.
• The term "±" indicates two possible solutions.
• The expression under the square root, \( b^2 - 4ac \), is called the discriminant, which helps us determine the nature of the roots.

Understanding the quadratic formula is crucial as it not only finds the solutions of the equation but also reveals deeper insights into the quadratic function's behavior. This formula is especially helpful when the quadratic does not factor neatly or if you're verifying solutions found through other methods.

Discriminant

The discriminant is an essential part of the quadratic formula that provides information about the nature of the roots of a quadratic equation. The discriminant is calculated as:\[ b^2 - 4ac. \]This value can tell us several important things about a quadratic equation based on whether it is positive, zero, or negative:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), there are no real roots, but rather two complex roots.

In our exercise, the discriminant was calculated as 25, which is positive. This confirms the presence of two real and distinct solutions to the quadratic equation. Understanding and calculating the discriminant is important as it quickly informs us of what to expect before solving the equation.

Graphing Quadratics

Graphing quadratic equations is a visual way to confirm the solutions obtained through algebraic methods, such as the quadratic formula. A quadratic equation graphs as a curve called a parabola, which can open upwards or downwards depending on the sign of the coefficient \( a \).To graph a quadratic equation effectively, consider the following steps:

• Find the vertex, which is the turning point of the parabola. The vertex can be found using the formula \( x = -\frac{b}{2a} \).
• Locate the axis of symmetry, which is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• Plot the roots or zeros of the quadratic (if they are real), which are the points where the parabola intersects the x-axis.
• Determine additional points on the parabola to get an accurate graph.

In the exercise provided, graphing confirmed that the solutions \( x = -\frac{1}{3} \) and \( x = -\frac{2}{3} \) correctly indicate where the parabola intersects the x-axis. Graphing not only verifies solutions but also aids in understanding the overall behavior of the quadratic function, such as its width, direction, and position.