Problem 22 - College Algebra

Question

Solve each equation in Exercises \(15-26\) by the square root method. $$(4 x-1)^{2}=16$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{5}{4}, \frac{-3}{4}\)

Step 1: Identify the Equation

First of all, rewrite the equation into the readable format: \( (4x-1)^2 = 16 \)

Step 2: Apply Square Root Method

In the square root method, take the square root of both sides of the equation. But remember, when dealing with square roots, there are always two options: positive and negative. That's why we get \(4x - 1 = \pm \sqrt{16}\), which simplifies to \(4x - 1 = \pm 4\)

Step 3: Solve for x

Next, move -1 to the right side of the equation to isolate \(x\). Therefore, we have:For the positive root:\(4x = 4 + 1 \rightarrow 4x = 5 \rightarrow x = \frac{5}{4}\)For the negative root: \(4x = -4 + 1 \rightarrow 4x = -3 \rightarrow x = \frac{-3}{4}\)

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra. These are polynomial equations of degree two, typically written in the form \(ax^2 + bx + c = 0\). In this form, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown variable we need to find. Quadratics are special because they form a parabola when graphed in a coordinate plane.
One important feature of quadratic equations is that they can have up to two real solutions. This characteristic is because their highest power is 2.
Quadratic equations appear in various contexts, from physics problems involving projectile motion to financial calculations predicting profit margins. Understanding how to solve them is thus a fundamental skill in mathematics.

Solving Equations

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this context, we focus on the square root method.
The square root method is especially useful for equations that can be simplified to the form \((px + q)^2 = r\). Here's a simplified version of how it works:

• Take the square root of both sides. Remember to include both the positive and negative roots due to the properties of square roots.
• Simplify the resulting linear equations. When you take the square root, the equation consists of a linear term, making it easier to solve.
• Isolate the variable by moving terms across the equation.

This method is efficient because it cuts directly to the solving stage by eliminating the quadratic term initially, provided the equation is in the right form.

Equation Solutions

Finding solutions to equations, especially quadratic ones, involves determining the values of \(x\) that satisfy the equation. In many cases, these solutions are real numbers, but quadratics can also have complex solutions, depending on the discriminant \(b^2 - 4ac\).
When using the square root method, as seen in the example \((4x - 1)^2 = 16\), we:

• Take the square root of both sides, which gives us \(4x - 1 = \pm 4\).
• Then solve the resulting linear equations to find two possible solutions: \(x = \frac{5}{4}\) and \(x = \frac{-3}{4}\).

These solutions can be quickly verified by substituting back into the original equation to check if they hold true, thus confirming their correctness. Solving equations is all about finding these values, understanding their implications, and verifying that they meet the conditions of the problem.