Problem 59 - College Algebra

Question

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. $$ \log _{11}\left(-2 x^{2}-7 x\right)=\log _{11}(x-2) $$

Step-by-Step Solution

Answer

The solution is \(x = -2 + \sqrt{5}\).

Step 1: Set the Arguments Equal to Each Other

Since the bases of the logarithms on both sides are the same (base 11), we can set the arguments equal to each other according to the property of logarithms. Thus, we have:\[ -2x^2 - 7x = x - 2 \]

Step 2: Rearrange the Equation

Let's move all terms to one side of the equation to form a quadratic equation:\[ -2x^2 - 7x - x + 2 = 0 \]This simplifies to:\[ -2x^2 - 8x + 2 = 0 \]

Step 3: Multiply by -1 to Simplify

To make calculations easier, multiply the entire equation by -1:\[ 2x^2 + 8x - 2 = 0 \]

Step 4: Solve the Quadratic Equation

Use the quadratic formula to solve for \(x\). The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \(a = 2\), \(b = 8\), and \(c = -2\). Substitute these values into the formula:\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \times 2 \times (-2)}}{2 \times 2} \]\[ x = \frac{-8 \pm \sqrt{64 + 16}}{4} \]\[ x = \frac{-8 \pm \sqrt{80}}{4} \]\[ x = \frac{-8 \pm 4 \sqrt{5}}{4} \]\[ x = -2 \pm \sqrt{5} \]

Step 5: Verify Solutions in the Original Equation

Check if both potential solutions \( x = -2 + \sqrt{5} \) and \( x = -2 - \sqrt{5} \) are valid in the context of the original log equation. Since \( \log_{11}(x-2) \) must be defined (its argument must be positive):- For \( x = -2 + \sqrt{5} \), \( -2 + \sqrt{5} - 2 \) must be positive.- For \( x = -2 - \sqrt{5} \), \( -2 - \sqrt{5} - 2 \) is negative, so it is not valid.Only \( x = -2 + \sqrt{5} \) is valid.

Step 6: Graph and Verify the Solution

Graph the functions \( y = \log_{11}(-2x^2 - 7x) \) and \( y = \log_{11}(x - 2) \). Identify the intersection point to verify the solution:Upon graphing, you should observe that the functions intersect at the point where the solution \( x = -2 + \sqrt{5} \) is valid, verifying our solution.

Key Concepts

Quadratic Equations

Quadratic equations represent mathematical expressions where the highest exponent of the variable is two. They often take the form \( ax^2 + bx + c = 0 \). Solving these equations usually involves finding the roots, or the values of \( x \) that satisfy the equation, which can be achieved using various methods such as factoring, completing the square, or utilizing the quadratic formula. The solution for the quadratic equation is given by the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this formula, \( a \), \( b \), and \( c \) are constants that come from the original quadratic equation. The term under the square root, \( b^2 - 4ac \), is known as the discriminant and determines the nature of the roots:

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

Understanding how to solve quadratic equations is crucial in various mathematical contexts, as these equations frequently appear in algebra, calculus, and even real-world applications such as physics and engineering.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are expressed in the form \( \, y = \log_b(x) \, \), where \( b \) is the base and \( x \) is the argument. The logarithm answers the question: 'To what power must \( b \) be raised, to get \( x \)?' Understanding the properties of logarithms is fundamental to solving logarithmic equations. Some key properties include:

• If \( \, \log_b(m) = \log_b(n) \, \), then \( \, m = n \).
• \( \, \log_b(m \,n) = \log_b(m) + \log_b(n) \).
• \( \, \log_b(m/n) = \log_b(m) - \log_b(n) \).
• \( \, \log_b(m^n) = n \, \log_b(m) \).

In solving the original logarithmic equation, knowing that the logs on both sides had the same base allowed us to set the arguments equal, facilitating the transformation into a quadratic equation. Thus, mastery of logarithmic rules and transformations equips students to tackle equations involving logs efficiently.

Graphing Techniques

Graphing provides a visual approach to solving equations and verifying solutions. By graphing equations, students can find intersection points, which represent solutions to the initial equations. For complex equations like logarithmic ones, graphing both sides of an equation allows evaluation of their behavior and the points where they coincide. To graph a function accurately, consider these steps:

• Identify key points such as intercepts and asymptotes.
• Determine the domain and range, especially for logarithmic functions where \( x > 0 \).
• Calculate values for different inputs to plot points.
• Connect the plotted points smoothly, respecting the nature of the function.

In the exercise at hand, graphing \( y = \log_{11}(-2x^2 - 7x) \) and \( y = \log_{11}(x - 2) \) was instrumental to verify the solution. This approach visually confirmed that the valid solution found, \( x = -2 + \sqrt{5} \), is where both graphs intersect. Graphing helps reinforce algebraic solutions by providing an intuitive, visual confirmation.