Solving (x+3)^2=36: A Step-by-Step Guide

Hey guys! Let's break down how to solve the quadratic equation $(x+3)^2 = 36$ using square roots. It's actually a pretty straightforward process once you get the hang of it. We'll go through each step in detail, so you can tackle similar problems with confidence.

Understanding Quadratic Equations

Before diving into the specifics of our problem, let's quickly recap what quadratic equations are and why square roots are handy for solving certain types. A quadratic equation is a polynomial equation of the second degree. The most general form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we're trying to find. However, the equation $(x+3)^2 = 36$ is a special case because it's in a form that allows us to directly apply the square root property.

The square root property states that if $x^2 = k$, then $x = \pm \sqrt{k}$. This property is super useful when the quadratic equation can be expressed as something squared equals a constant, which is exactly what we have in our case. Using square roots simplifies the process and avoids the need for factoring or using the quadratic formula in this particular scenario. It's all about choosing the right tool for the job to make things as easy as possible!

Step-by-Step Solution

Okay, let's get into the nitty-gritty and solve $(x+3)^2 = 36$ step-by-step. Trust me, it's easier than it looks!

Step 1: Take the Square Root of Both Sides

The first thing we need to do is get rid of that square on the left side. To do this, we take the square root of both sides of the equation. Remember, what you do to one side, you have to do to the other to keep the equation balanced.

So, we have:

$\sqrt{(x+3)^2} = \pm \sqrt{36}$

Notice the $\pm$ sign on the right side. This is super important because both the positive and negative square roots of 36 will satisfy the original equation. For example, both $6^2$ and $(-6)^2$ equal 36.

Step 2: Simplify

Now, let's simplify both sides. The square root of $(x+3)^2$ is simply $(x+3)$, and the square root of 36 is 6. So, we now have:

$x + 3 = \pm 6$

This gives us two separate equations to solve:

1. $x + 3 = 6$
2. $x + 3 = -6$

Step 3: Solve for x

Now, we just need to isolate $x$ in each of these equations. To do that, we subtract 3 from both sides of each equation.

For the first equation:

$x + 3 - 3 = 6 - 3$

$x = 3$

For the second equation:

$x + 3 - 3 = -6 - 3$

$x = -9$

So, we have two solutions for $x$: $x = 3$ and $x = -9$.

Step 4: Check Your Solutions

It's always a good idea to check your solutions to make sure they work in the original equation. This helps prevent errors and ensures you've got the right answers.

Let's check $x = 3$:

$(3 + 3)^2 = 36$

$6^2 = 36$

$36 = 36$ (This checks out!)

Now, let's check $x = -9$:

$(-9 + 3)^2 = 36$

$(-6)^2 = 36$

$36 = 36$ (This also checks out!)

Both solutions work, so we're good to go!

Alternative Methods

While using square roots is the most direct method for solving this particular equation, it's worth mentioning a couple of other ways you could approach it. These methods might be useful in different situations or when you're dealing with more complex quadratic equations.

Expanding and Factoring

One alternative is to expand the left side of the equation, rewrite it in standard quadratic form, and then try to factor it. Let's see how that would work:

$(x + 3)^2 = 36$

Expand the left side:

$x^2 + 6x + 9 = 36$

Subtract 36 from both sides to set the equation to zero:

$x^2 + 6x - 27 = 0$

Now, we need to factor the quadratic expression. We're looking for two numbers that multiply to -27 and add to 6. Those numbers are 9 and -3. So, we can factor the equation as:

$(x + 9)(x - 3) = 0$

Setting each factor equal to zero gives us:

$x + 9 = 0$ or $x - 3 = 0$

Solving for $x$ gives us:

$x = -9$ or $x = 3$

As you can see, we get the same solutions as before, but this method involves a bit more algebra.

Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation, regardless of its form. The formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our equation, $x^2 + 6x - 27 = 0$, we have $a = 1$, $b = 6$, and $c = -27$. Plugging these values into the quadratic formula gives us:

$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-27)}}{2(1)}$

$x = \frac{-6 \pm \sqrt{36 + 108}}{2}$

$x = \frac{-6 \pm \sqrt{144}}{2}$

$x = \frac{-6 \pm 12}{2}$

This gives us two solutions:

$x = \frac{-6 + 12}{2} = \frac{6}{2} = 3$

$x = \frac{-6 - 12}{2} = \frac{-18}{2} = -9$

Again, we arrive at the same solutions: $x = 3$ and $x = -9$. The quadratic formula is a bit more involved, but it's a reliable method when factoring isn't straightforward or when you just want a guaranteed solution.

Real-World Applications

You might be wondering, "Where would I ever use this stuff in real life?" Well, quadratic equations pop up in various fields, including physics, engineering, and even finance. Here are a couple of examples:

Projectile Motion

In physics, quadratic equations are used to describe the trajectory of a projectile, like a ball thrown into the air. The equation can help you determine how high the ball will go and how far it will travel before hitting the ground. Understanding these equations is crucial for designing things like sports equipment or even planning artillery fire (though hopefully, you won't need to do that!).

Engineering Design

Engineers use quadratic equations to design structures like bridges and buildings. These equations help calculate the forces and stresses acting on different parts of the structure, ensuring that it can withstand those forces and remain stable. For example, when designing an arch, engineers need to consider the quadratic equation that describes the shape of the arch to ensure it can support the load above it.

Financial Modeling

In finance, quadratic equations can be used to model various scenarios, such as the growth of an investment or the depreciation of an asset. These models can help investors make informed decisions about where to put their money and how to manage their risk. For example, a quadratic equation might be used to model the relationship between the price of a stock and the demand for that stock.

Tips and Tricks

Here are some handy tips and tricks to keep in mind when solving quadratic equations:

• Always check for simplifications: Before diving into any method, see if you can simplify the equation. In our case, recognizing that $(x+3)^2 = 36$ is a perfect square made the problem much easier to solve.
• Don't forget the $\pm$: When taking the square root of both sides of an equation, always remember to include both the positive and negative roots. This ensures you find all possible solutions.
• Check your solutions: Plugging your solutions back into the original equation is a great way to catch errors and ensure you've got the correct answers.
• Know your methods: Familiarize yourself with different methods for solving quadratic equations, such as factoring, the quadratic formula, and completing the square. This will give you more flexibility when tackling different types of problems.
• Practice, practice, practice: The more you practice solving quadratic equations, the more comfortable you'll become with the process. Try working through a variety of problems to build your skills and confidence.

Conclusion

So, there you have it! Solving the quadratic equation $(x+3)^2 = 36$ using square roots is a straightforward process once you understand the basic steps. Remember to take the square root of both sides, consider both positive and negative roots, and check your solutions. And don't be afraid to explore alternative methods like factoring or the quadratic formula. With a little practice, you'll be solving quadratic equations like a pro in no time!