Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Which one of the following equations is set up for direct use of the square root property? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Step-by-Step Solution

Answer

Equation B is set up for the square root property, leading to solutions \(x = \frac{\sqrt{7} - 5}{2}\) and \(x = \frac{-\sqrt{7} - 5}{2}\).

Step 1: Identify the Square Root Property Equation

The square root property can be applied directly to equations of the form \(a(x-h)^2 = k\), where you can solve by taking the square root of both sides after isolating the squared term. Among the given options, the equation \((2x+5)^2=7\) belongs to this form.

Step 2: Isolate the Square and Take the Square Root

Since the equation is already in the form \((2x+5)^2 = 7\), we can directly apply the square root property by taking the square root of both sides of the equation: \[ (2x+5) = \pm \sqrt{7}. \]

Step 3: Solve for x

To solve for \(x\), isolate \(x\) in each of the cases from the previous step:1. For the first case, \(2x+5 = \sqrt{7}\): - Subtract 5 from both sides: \[2x = \sqrt{7} - 5.\] - Divide by 2: \[x = \frac{\sqrt{7} - 5}{2}.\]2. For the second case, \(2x+5 = -\sqrt{7}\): - Subtract 5 from both sides: \[2x = -\sqrt{7} - 5.\] - Divide by 2: \[x = \frac{-\sqrt{7} - 5}{2}.\]

Key Concepts

Quadratic Equations

Quadratic equations are a foundational component in algebra and precalculus. They are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The highest power of \(x\) is 2, which gives the equation its "quadratic" nature. These equations can graph as parabolas and can have two, one, or no real solutions.
When studying quadratic equations, we often encounter different methods to solve them, such as factoring, completing the square, using the quadratic formula, or applying the square root property. Each method has certain forms of equations they work best for.

• Factoring: Works well when the quadratic easily breaks down into two binomial factors.
• Completing the Square: Involves modifying the equation to make it a perfect square trinomial.
• Quadratic Formula: A universal method that can solve any quadratic equation.
• Square Root Property: Effective when the equation is in the form \((x-h)^2 = k\).

Understanding these basic concepts provides the foundation needed for mastering quadratic equations and their solutions.

Solving Equations

Solving equations is the process of finding the value(s) of the unknown variable that make an equation true. In the context of quadratic equations, it refers to the methods and steps used to determine the solutions to equations of the form \(ax^2 + bx + c = 0\).
When using the square root property, you work with equations that have an isolated squared term. The square root property allows you to take the square root of both sides of an equation when it is structured as \((x-h)^2 = k\). This method is particularly straightforward, as it involves only a few steps:

• Isolate the squared term.
• Apply the square root to both sides of the equation.
• Solve for the variable by isolating it on one side.

For example, with the equation \((2x+5)^2 = 7\), apply the square root property by taking the square root of both sides to find \(2x + 5 = \pm \sqrt{7}\), then solve for \(x\) accordingly. Remember, when taking the square root of both sides, always consider the positive and negative solutions.

Precalculus Methods

Precalculus methods encompass a variety of mathematical techniques and strategies used to analyze and solve more complex equations beyond simple algebra. These methods bridge the gap between algebra and calculus, focusing on strengthening problem-solving skills and understanding function behavior. Using the square root property is one such technique that appears in precalculus studies.
In the specific context of solving quadratic equations, precalculus methods aim to make problem solving more efficient by identifying the most appropriate technique for a given problem. When an equation fits a particular form, like the square root suitable form \((x-h)^2 = k\), we use precalculus wisdom to employ the square root property efficiently. This approach not only simplifies the calculations but also enhances your conceptual understanding of the equation's behavior.

• Efficiency: Identifying and applying the most suitable method saves time and reduces complexity.
• Understanding: Deepens comprehension of function properties and equation manipulations.
• Anticipation: Prepares students for calculus by fostering a robust understanding of foundational concepts.

Learning and applying these methods is vital for students as they advance mathematically, allowing them to solve complex problems with confidence and ease.