Solve (8x+5)^2=7 Using The Square Root Property

In the realm of mathematics, particularly in algebra, the square root property stands as a pivotal technique for solving certain types of equations. This method proves especially useful when dealing with equations where a variable expression is squared. In this comprehensive guide, we will delve into the intricacies of the square root property, elucidating its principles, demonstrating its application through a detailed example, and underscoring its significance in solving quadratic equations. Our primary focus will be on resolving the equation $(8x + 5)^2 = 7$ using this powerful property. We will meticulously walk through each step, ensuring clarity and comprehension, and finally, we will present the simplified solution, adhering to the required format of exact answers with radicals, separated by commas.

Understanding the Square Root Property

The square root property is a fundamental concept in algebra that provides a direct method for solving equations in the form of $u^2 = c$, where $u$ is an algebraic expression and $c$ is a constant. The essence of this property lies in the understanding that if the square of a quantity equals a certain value, then that quantity must be equal to either the positive or the negative square root of that value. Mathematically, this can be expressed as follows:

If $u^2 = c$, then $u = \pm \sqrt{c}$

This property stems from the basic principle that squaring both a positive and a negative number yields a positive result. For instance, both $3^2$ and $(-3)^2$ equal 9. Therefore, when we encounter an equation like $x^2 = 9$, we must consider both the positive and negative square roots of 9, which are 3 and -3, respectively. The square root property is not merely a mathematical trick; it is a logical consequence of the definition of square roots and squares.

It's important to note that the constant $c$ plays a crucial role in determining the nature of the solutions. If $c$ is a positive number, then the equation will have two distinct real solutions, one positive and one negative. If $c$ is zero, the equation will have one real solution, which is zero itself. If $c$ is a negative number, the equation will have two complex solutions, involving the imaginary unit $i$, where $i = \sqrt{-1}$. Understanding these nuances is vital for the correct application of the square root property in various scenarios. In our specific case, we are dealing with a positive constant, which means we will obtain two distinct real solutions.

Applying the Square Root Property to $(8x + 5)^2 = 7$

Now, let's apply the square root property to solve the given equation: $(8x + 5)^2 = 7$. This equation is in the form $u^2 = c$, where $u = (8x + 5)$ and $c = 7$. According to the square root property, if $(8x + 5)^2 = 7$, then $(8x + 5)$ must be equal to either the positive or negative square root of 7. This can be written as:

$8x + 5 = \pm \sqrt{7}$

This equation now branches into two separate equations, one for the positive square root and one for the negative square root:

1. $8x + 5 = \sqrt{7}$
2. $8x + 5 = -\sqrt{7}$

We will solve each of these equations individually to find the two possible values of $x$. For the first equation, $8x + 5 = \sqrt{7}$, we need to isolate $x$. The first step is to subtract 5 from both sides of the equation:

$8x = \sqrt{7} - 5$

Next, we divide both sides by 8 to solve for $x$:

$x = \frac{\sqrt{7} - 5}{8}$

This gives us our first solution. Now, let's solve the second equation, $8x + 5 = -\sqrt{7}$. Again, we start by subtracting 5 from both sides:

$8x = -\sqrt{7} - 5$

Then, we divide both sides by 8 to isolate $x$:

$x = \frac{-\sqrt{7} - 5}{8}$

This gives us our second solution. Thus, we have found two distinct solutions for $x$ by applying the square root property. These solutions are exact and involve radicals, as required.

Simplifying and Presenting the Solution

The solutions we obtained in the previous section are:

1. $x = \frac{\sqrt{7} - 5}{8}$
2. $x = \frac{-\sqrt{7} - 5}{8}$

These solutions are already in their simplest form, as the radical $\sqrt{7}$ cannot be simplified further, and the fractions are reduced to their lowest terms. The solutions are also exact, meaning they are not approximations obtained through decimal calculations. Exact solutions are crucial in many mathematical contexts, especially when dealing with irrational numbers like $\sqrt{7}$, as they preserve the precise value without introducing rounding errors.

To present the solutions in the requested format, we need to separate them by a comma. Therefore, the final answer is:

$x = \frac{\sqrt{7} - 5}{8}, \frac{-\sqrt{7} - 5}{8}$

This format clearly presents both solutions, adhering to the instructions provided. The solutions represent the two values of $x$ that satisfy the original equation $(8x + 5)^2 = 7$. To verify these solutions, one could substitute each value back into the original equation and confirm that both sides of the equation are equal. This process of verification is a good practice to ensure the accuracy of the solutions, especially when dealing with more complex equations.

Importance of the Square Root Property

The square root property is not merely a technique for solving specific types of equations; it is a fundamental tool in the broader context of algebra and equation solving. Its importance stems from its ability to directly address equations where a variable expression is squared, which frequently arise in various mathematical problems. Without the square root property, solving such equations would require more complex methods, such as expanding the squared expression and then employing the quadratic formula or factoring techniques.

One of the primary advantages of the square root property is its efficiency. When applicable, it provides a straightforward and direct path to the solutions, avoiding the often cumbersome steps involved in other methods. This efficiency is particularly valuable in situations where time is a constraint, such as in examinations or timed problem-solving scenarios. Moreover, the square root property offers a clear and intuitive way to understand the nature of solutions to quadratic equations, highlighting the concept of two possible roots (positive and negative) arising from the square root operation.

Furthermore, the square root property serves as a building block for understanding more advanced concepts in algebra and calculus. Its principles are used in solving more complex equations, in simplifying expressions involving radicals, and in various applications in geometry and physics. For example, the Pythagorean theorem, a cornerstone of geometry, often leads to equations that can be solved using the square root property. In physics, problems involving motion and energy frequently result in equations where the square root property is an essential tool.

In addition to its practical applications, the square root property also reinforces the fundamental principles of algebraic manipulation and equation solving. It underscores the importance of performing the same operation on both sides of an equation to maintain equality and highlights the inverse relationship between squaring and taking the square root. This understanding is crucial for developing a strong foundation in algebra and for tackling more advanced mathematical challenges.

In conclusion, the square root property is a powerful and essential tool in the arsenal of any mathematics student or practitioner. Its efficiency, clarity, and broad applicability make it a valuable asset in solving equations and understanding the fundamental concepts of algebra. Mastering this property is not only crucial for success in mathematics courses but also for tackling real-world problems that involve mathematical modeling and analysis.

Repair Input Keyword

Solve the equation $(8x + 5)^2 = 7$ for $x$ using the square root property. Simplify your answer and express it in exact form, using radicals as needed. Separate multiple answers with a comma.

SEO Title

Solving Equations with Square Root Property: A Step-by-Step Guide 📚