Solve $x^2 + 10x = 24$ By Completing The Square

Hey guys! Today, we're diving into the world of quadratic equations and tackling a specific problem using a cool technique called "completing the square." We'll break down the steps to solve the equation $x^2 + 10x = 24$, and by the end, you'll not only have the answer but also understand why this method works. So, let's get started!

Understanding the Problem: $x^2 + 10x = 24$

Before we jump into the solution, let's make sure we understand what we're dealing with. Quadratic equations are equations where the highest power of the variable (in this case, x) is 2. They often have the general form $ax^2 + bx + c = 0$, where a, b, and c are constants. Our equation, $x^2 + 10x = 24$, fits this description, although it's not quite in the standard form yet because it doesn't equal zero on one side. The goal here is to find the values of x that make this equation true. These values are called the solutions or roots of the equation.

In this equation, we have $x^2$ which is the quadratic term, $10x$ which is the linear term, and 24 which is a constant. To solve this by completing the square, we'll manipulate the equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. This method is particularly useful when the quadratic equation doesn't factor easily, or when we need to rewrite the equation in vertex form. You might be wondering why completing the square is even important when we have other methods like the quadratic formula. Well, mastering this technique gives you a deeper understanding of quadratic equations and their properties. It also lays the groundwork for understanding more advanced mathematical concepts. So, stick with me, and you'll see how powerful this method can be!

Step 1: Move the Constant Term

The first thing we need to do is move the constant term (24) to the right side of the equation. We already have it there, so in this specific case, we can skip this step! However, if our equation was something like $x^2 + 10x + 5 = 0$, we would subtract 5 from both sides to get $x^2 + 10x = -5$. This step ensures that we have only the $x^2$ and $x$ terms on the left side, which is crucial for completing the square.

So, let’s just reiterate our starting point: we have $x^2 + 10x = 24$. We’re all set to move on to the next step, where the real magic begins!

Step 2: Completing the Square

This is where the heart of the method lies. To complete the square, we need to add a specific constant to both sides of the equation. This constant will transform the left side into a perfect square trinomial. The question is, how do we find this magical number? Well, here's the trick:

1. Take half of the coefficient of the x term. In our equation, the coefficient of the x term is 10. Half of 10 is 5.
2. Square the result from the previous step. 5 squared ($5^2$) is 25.

This number, 25, is what we need to add to both sides of the equation. By adding 25, we're essentially forcing the left side to become a perfect square. So, let's do it:

$x^2 + 10x + 25 = 24 + 25$

This simplifies to:

$x^2 + 10x + 25 = 49$

Now, the left side is a perfect square trinomial. Let's see why this is so important in the next step.

Step 3: Factor the Perfect Square Trinomial

Remember what a perfect square trinomial is? It's a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. Our left side, $x^2 + 10x + 25$, is indeed a perfect square trinomial. It factors beautifully into:

$(x + 5)^2$

Why $(x + 5)^2$? Think about it: when you expand $(x + 5)(x + 5)$, you get $x^2 + 5x + 5x + 25$, which simplifies to $x^2 + 10x + 25$. So, we've successfully rewritten the left side of our equation in a much more compact and useful form.

Now, our equation looks like this:

$(x + 5)^2 = 49$

We're getting closer to the solution! We've managed to isolate the variable x within a squared term. The next step will involve undoing that square.

Step 4: Take the Square Root of Both Sides

To get rid of the square on the left side, we need to take the square root of both sides of the equation. This is a crucial step, but we need to be careful! Remember that when you take the square root of a number, there are actually two possible solutions: a positive one and a negative one.

So, when we take the square root of both sides of $(x + 5)^2 = 49$, we get:

$x + 5 = ±

Why ±7? Because both 7 squared (7 * 7) and -7 squared (-7 * -7) equal 49. This is why it's super important to include both the positive and negative roots when solving quadratic equations.

Now we have two separate equations to solve:

1. $x + 5 = 7$
2. $x + 5 = -7$

Let's tackle these one at a time in the next step.

Step 5: Solve for x

We now have two simple linear equations to solve for x. Let's start with the first one:

$x + 5 = 7$

To isolate x, we subtract 5 from both sides:

$x = 7 - 5$

$x = 2$

So, one solution is x = 2.

Now, let's solve the second equation:

$x + 5 = -7$

Again, subtract 5 from both sides:

$x = -7 - 5$

$x = -12$

So, our second solution is x = -12.

We've done it! We've found the two values of x that satisfy the original equation.

Step 6: Express the Solution Set

The final step is to express our solutions as a set. The solution set is simply a list of all the solutions, enclosed in curly braces. In our case, the solutions are 2 and -12, so the solution set is:

$\{-12, 2\}$

This means that if we substitute either -12 or 2 for x in the original equation, $x^2 + 10x = 24$, the equation will be true. We can even check this to be sure!

Checking the Solutions

Let's check our solutions to make sure they work. First, let's check x = 2:

$(2)^2 + 10(2) = 24$

$4 + 20 = 24$

$24 = 24$

Yep, x = 2 works!

Now, let's check x = -12:

$(-12)^2 + 10(-12) = 24$

$144 - 120 = 24$

$24 = 24$

Awesome, x = -12 works too!

We've successfully verified that both solutions are correct.

Conclusion

So, there you have it! We've solved the quadratic equation $x^2 + 10x = 24$ by completing the square. We walked through each step, from moving the constant term to expressing the solution set. Remember, the key to completing the square is to add the right constant to both sides of the equation to create a perfect square trinomial. This allows you to factor the equation, take the square root of both sides, and solve for x.

Completing the square might seem a bit tricky at first, but with practice, it becomes a powerful tool in your mathematical arsenal. It not only helps you solve quadratic equations but also provides a deeper understanding of their structure and properties. Keep practicing, and you'll master it in no time!

The solution set of the equation $x^2 + 10x = 24$ is C. $\{-12, 2\}$. We found this by completing the square, factoring, taking square roots, and solving the resulting linear equations. Remember to always check your solutions to ensure they are correct.

Keep up the great work, and I'll see you in the next math adventure!