Problem 22 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(4 x+1)^{2}=20$$

Step-by-Step Solution

Answer

The solutions are \(x \approx 0.7808\) and \(x \approx -1.2808\), confirmed graphically.

Step 1: Understand the Equation

The given equation is \((4x+1)^{2} = 20\). It involves a squared binomial, which means we will need to isolate the binomial and then solve for \(x\).

Step 2: Take the Square Root of Both Sides

To eliminate the square, take the square root of both sides: \((4x+1) = \sqrt{20}\) or \((4x+1) = -\sqrt{20}\). The square root of 20 can be simplified to \(2\sqrt{5}\). So, the two equations we have are: \((4x+1) = 2\sqrt{5}\) and \((4x+1) = -2\sqrt{5}\).

Step 3: Solve for x

From \(4x + 1 = 2\sqrt{5}\), subtract 1 from both sides: \(4x = 2\sqrt{5} - 1\). Then divide by 4: \(x = \frac{2\sqrt{5} - 1}{4}\). From \(4x + 1 = -2\sqrt{5}\), similarly, solve to get \(x = \frac{-2\sqrt{5} - 1}{4}\).

Step 4: Graphical Explanation

Plotting the functions \(y = (4x+1)^{2}\) and \(y = 20\) on a graph will show their intersection points. These points represent the solutions we found: \(x = \frac{2\sqrt{5} - 1}{4}\) and \(x = \frac{-2\sqrt{5} - 1}{4}\).

Step 5: Simplify and Conclude

Simplify and approximate \(\sqrt{5} \approx 2.236\), so solutions are approximately \(x \approx 0.7808\) and \(x \approx -1.2808\). These are real solutions confirmed by the graph.

Key Concepts

Square Roots

When working with equations that involve square roots, it's important to understand their role in solving the problem. A square root essentially helps us find a number which, when multiplied by itself, equals the original number under the square root symbol. In the equation
\[(4x+1)^2=20\],
taking the square root of both sides is crucial. This operation helps us eliminate the square and brings the equation to a simpler form. For instance, when we take the square root of both sides in the expression:

• \((4x + 1) = \sqrt{20}\)
• or \((4x + 1) = -\sqrt{20}\)

The square root of 20 is further simplified into \(2\sqrt{5}\). It's important to note that square roots have both positive and negative solutions, hence the plus-minus consideration. Breaking down complex numbers into their square roots can make solving equations more manageable.

Graphical Solutions

A graphical solution is a visual method of finding where two mathematical statements equal each other. We achieve this by plotting both equations on a graph and determining their intersection points. For the given equation:
\((4x+1)^{2}=20\),
we plot two separate functions:

• \(y = (4x+1)^{2}\)
• \(y = 20\)

These two graphs will intersect at points that correspond to the values of \(x\) which satisfy the equation. These intersection points visually verify the solutions found algebraically. This method is particularly helpful when dealing with real solutions, as it provides a direct and visual confirmation of the work. Graphical solutions enhance our understanding of the behavior of functions and their relationships.

Binomial Expressions

Binomial expressions are algebraic expressions that contain two terms connected by either addition or subtraction. In the equation
\((4x+1)^2=20\),
the expression \(4x+1\) is a binomial. Squaring a binomial expression leads to expanded forms that we often need to condense while solving equations.When solving binomials that are squared, the key process usually begins with eliminating the square—this was done by taking the square root in our problem. After simplifying the equation, we isolate the variable term. Understanding how to manipulate these binomials by expanding, factoring, or simplifying them is essential in algebra.Handling binomial expressions allows solving a wide range of algebraic problems and this knowledge is fundamental to progressing in more complex topics within algebra.