Solve X² - 36 = 0: Simple Quadratic Equation

Welcome, math enthusiasts! Today, we're diving into a common type of algebraic problem: solving a quadratic equation. Specifically, we'll be tackling the equation $x^2 - 36 = 0$. This is a fantastic example that illustrates a fundamental concept in algebra, and understanding how to solve it will pave the way for more complex equations. We'll break down the steps, explore different methods, and arrive at the correct solution. So, grab your thinking caps, and let's get started on this journey to solve the quadratic equation $x^2 - 36 = 0$ and master the art of finding unknown values.

Understanding Quadratic Equations

Before we jump into solving $x^2 - 36 = 0$, let's take a moment to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term that is squared. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our specific case, $x^2 - 36 = 0$, we can see that it fits this general form. Here, $a = 1$ (because $x^2$ is the same as $1x^2$), $b = 0$ (because there is no 'x' term), and $c = -36$. The 'b = 0' is what makes this equation particularly straightforward to solve. The goal when we solve the quadratic equation is to find the value(s) of 'x' that make the equation true. These values are called the roots or solutions of the equation. For quadratic equations, there can be zero, one, or two real solutions.

Method 1: Isolating the Variable

One of the most intuitive ways to solve the quadratic equation $x^2 - 36 = 0$ is by isolating the $x^2$ term. This method works best when the 'bx' term is zero, as it is in our equation. Here's how we do it:

1. Start with the equation: $x^2 - 36 = 0$
2. Add 36 to both sides: Our aim is to get $x^2$ by itself on one side of the equation. To do this, we need to eliminate the '-36'. We can achieve this by adding 36 to both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other to maintain equality. $x^2 - 36 + 36 = 0 + 36$ This simplifies to: $x^2 = 36$
3. Take the square root of both sides: Now that we have $x^2$ isolated, we need to find out what 'x' is. To undo the squaring of 'x', we take the square root of both sides of the equation. It's crucial to remember that when you take the square root of a number, there are always two possible results: a positive one and a negative one. For example, both $6^2$ (which is $6 imes 6$) and $(-6)^2$ (which is $-6 imes -6$) equal 36. $\sqrt{x^2} = \sqrt{36}$ This gives us: $x = \pm 6$

Therefore, the solutions for 'x' are $x = 6$ and $x = -6$. This method is efficient and direct for equations of this specific form, making it a go-to technique when the 'bx' term is absent.

Method 2: Factoring the Difference of Squares

Another powerful way to solve the quadratic equation $x^2 - 36 = 0$ is by using the factoring technique, specifically the 'difference of squares' formula. This is a very common algebraic identity that states: $a^2 - b^2 = (a - b)(a + b)$.

Let's apply this to our equation:

1. Recognize the pattern: Our equation, $x^2 - 36 = 0$, perfectly fits the difference of squares pattern. Here, $a^2$ is $x^2$ (so $a = x$) and $b^2$ is 36 (so $b = 6$, since $6^2 = 36$).
2. Apply the formula: Using the difference of squares formula, we can rewrite the equation as: $(x - 6)(x + 6) = 0$
3. Set each factor to zero: For the product of two factors to be zero, at least one of the factors must be zero. This is known as the Zero Product Property. So, we set each factor equal to zero and solve for 'x' in each case:
   • First factor: $x - 6 = 0$ Add 6 to both sides: $x = 6$
   • Second factor: $x + 6 = 0$ Subtract 6 from both sides: $x = -6$

As you can see, factoring also yields the same solutions: $x = 6$ and $x = -6$. This method is not only effective but also reinforces the understanding of algebraic identities and their applications in simplifying and solving equations. It's a skill that will serve you well as you encounter more complex polynomial equations.

Verifying the Solutions

It's always a good practice to verify the solutions we find to ensure they are correct. Let's plug our solutions, $x = 6$ and $x = -6$, back into the original equation $x^2 - 36 = 0$.

• For x = 6: $(6)^2 - 36 = 0$ $36 - 36 = 0$ $0 = 0$ This is true, so $x = 6$ is a correct solution.

• For x = -6: $(-6)^2 - 36 = 0$ $36 - 36 = 0$ $0 = 0$ This is also true, so $x = -6$ is a correct solution.

Both solutions satisfy the original equation, confirming that our methods were applied correctly and our answers are accurate. This verification step is a crucial part of the problem-solving process in mathematics, helping to build confidence in your results.

Conclusion

In conclusion, we've successfully learned how to solve the quadratic equation $x^2 - 36 = 0$ using two distinct and powerful methods: isolating the variable and factoring the difference of squares. Both approaches led us to the same correct solutions: $x = 6$ and $x = -6$. These methods highlight the elegance and versatility of algebraic techniques. Understanding how to manipulate equations and apply specific formulas like the difference of squares is fundamental to mastering algebra and tackling more advanced mathematical challenges. Remember, practice is key! The more you work through problems like this, the more comfortable and proficient you'll become.

For further exploration into quadratic equations and algebraic concepts, you might find the resources at Khan Academy very helpful. They offer a wealth of free educational materials, including detailed explanations and practice exercises on a wide range of mathematical topics, from basic algebra to calculus. Exploring their site can provide you with additional insights and support as you continue your mathematical journey.

Answer Choices:

A. $x=1 ; x=-36$ B. $x=-1 ; x=36$ C. $x=-6 ; x=6$ D. $x=-18 ; x=18

Based on our calculations, the correct answer is C. $x=-6 ; x=6$.