Understanding Solutions to Quadratic Equations with ACT Aspire Math

If x^2 = 36, what are the possible values of x?

• A. x = 6 or x = -6
• B. x = 6 or x = 0
• C. x = -6 or x = 0
• D. x = 12 or x = -12

Answer: (A)

The equation \( x^2 = 36 \) indicates that we are looking for values of \( x \) that, when squared, yield 36. To solve for \( x \), we can take the square root of both sides of the equation. When taking the square root, it's important to remember that both the positive and negative roots are valid solutions for \( x \). Therefore, we have: \[ x = \sqrt{36} \text{ which gives us } x = 6, \] and \[ x = -\sqrt{36} \text{ which gives us } x = -6. \] This means \( x \) can be either 6 or -6. These two values represent all possible solutions to the equation, as squaring either one will return 36. The other choices do not provide valid solutions based on the equation given. Specifically, zero cannot be a solution because squaring zero results in zero, not 36. Also, values like 12 and -12 do not satisfy the original equation when squared, as they yield 144, not 36. Thus, the correct answer identifying the possible values of \( x \) is indeed \( x = 6 \)

Have you ever come across an equation like ( x^2 = 36 ) and wondered what values ( x ) could possibly take? You're definitely not alone! The beauty of mathematics lies in its structure and predictability, especially when it comes to solving equations. So, let’s break this down and understand how to find the answers, particularly if you're preparing for the Mathematics ACT Aspire Practice Test.

First off, the equation ( x^2 = 36 ) is a basic quadratic equation. It tells us that ( x ), when squared, results in 36. Sounds simple enough, right? But here's where the magic (and sometimes confusion) happens: there are actually two values of ( x ) that will satisfy this equation. Yes, you heard that right—two!

What Are the Values?

To track down these elusive values, we take the square root of both sides. But here’s a little nugget of wisdom—when you take the square root, you need to consider both the positive and negative roots. So, we can express it simply as:

[

x = \sqrt{36} \quad \text{(which gives us)} \quad x = 6

]

And then there’s:

[

x = -\sqrt{36} \quad \text{(which gives us)} \quad x = -6

]

So, voilà! The possible values of ( x ) are 6 and -6.

Why Not the Other Options?

Looking at the choices provided—A. ( x = 6 ) or ( x = -6 ), B. ( x = 6 ) or ( x = 0 ), C. ( x = -6 ) or ( x = 0 ), D. ( x = 12 ) or ( x = -12 )—it’s clear that only the first option holds true. Why? Because squaring 0 returns 0, not 36, and ( 12^2 ) or ( -12^2 ) gives us 144. Simple enough, wouldn’t you say?

Real Life Applications

Understanding quadratic equations is not just a preparation tool for tests—it's a key ingredient in many life situations. Ever tried to calculate the area of a square? Yep, you’re using squares. How about physics? Trajectories of balls thrown or kicked follow a parabolic path governed by quadratic equations. It might seem theoretical, but this knowledge extends into the real world.

And it's not just about equations—what about building confidence in math? Knowing how to tackle quadratic equations can set the groundwork for more complex topics like functions and graphing. So, while studying for your ACT Aspire Mathematics test, keep your focus sharp on understanding these principles rather than simply memorizing.

Mind the Gaps

If you ever find yourself lost, don’t hesitate to seek resources like textbooks, online tutorials, or peer study groups. They can offer additional perspectives on tricky concepts that might at first seem overwhelming.

So, the next time you encounter ( x^2 = 36 ), remember: it’s not just about crunching numbers—it's about embracing the journey of understanding logic through mathematics. And knowing that ( x = 6 ) or ( x = -6 ) is just a small step in a much larger world of numbers. Happy studying, and remember to have fun with it!