Solving Quadratic Equations In Disguise A Step-by-Step Guide For (2x + 6)^2 - 10(2x + 6) + 22 = -2

Hey guys! Today, we're diving into solving a quadratic equation that might look a bit intimidating at first glance. But don't worry, we'll break it down step by step and you'll see it's totally manageable. The equation we're tackling is:

(2x + 6)^2 - 10(2x + 6) + 22 = -2

This equation is in what we call quadratic form. What does that mean? Well, it means we can use a clever substitution to make it look like a standard quadratic equation, which we know how to solve. Think of it like this: we're going to give the complicated part of the equation a new, simpler name, solve for that name, and then go back and figure out what the original variable, x, is equal to.

Step 1: Simplify and Rearrange

Before we make our substitution, let's simplify the equation a bit and get it into a more standard form. Right now, we have a constant term (-2) on the right side of the equation. Let's move that over to the left side by adding 2 to both sides. This gives us:

(2x + 6)^2 - 10(2x + 6) + 22 + 2 = 0

Which simplifies to:

(2x + 6)^2 - 10(2x + 6) + 24 = 0

Now our equation looks a little cleaner and we're ready for the next step.

Step 2: Make a Substitution

This is the key step in solving equations in quadratic form. We're going to pick a new variable to represent the repeating expression in our equation. Notice that the expression (2x + 6) appears twice. Let's make the substitution:

Let y = (2x + 6)

Now we can replace every instance of (2x + 6) in our equation with y. This transforms our equation into:

y^2 - 10y + 24 = 0

Wow! Doesn't that look much more familiar? This is a standard quadratic equation in terms of y. We've successfully transformed our original equation into a simpler form that we can solve.

This step is really the heart of solving equations in quadratic form. By recognizing the repeating expression and making a substitution, we can turn a complex-looking equation into something we already know how to handle. It's like a magic trick, but with math!

Think of this substitution as a temporary change of clothes for the equation. We're dressing it up in a way that makes it easier to work with. Once we've solved for y, we can change it back into its original form and solve for x. It's all about making the problem more manageable.

So, with our substitution in place, we've gone from a somewhat intimidating equation to a friendly quadratic equation that's practically begging to be solved. Let's move on to the next step and find the values of y that satisfy this equation.

Step 3: Solve the Quadratic Equation for y

We now have a standard quadratic equation:

y^2 - 10y + 24 = 0

There are a couple of ways we can solve this. We can try factoring, or we can use the quadratic formula. In this case, factoring looks like it might be the easier route. We need to find two numbers that multiply to 24 and add up to -10. Can you think of what they are?

The numbers are -6 and -4, because (-6) * (-4) = 24 and (-6) + (-4) = -10. So, we can factor the quadratic equation as:

(y - 6)(y - 4) = 0

Now, we use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions for y:

• y - 6 = 0 => y = 6
• y - 4 = 0 => y = 4

So, we've found two values for y: 6 and 4. But remember, we're not trying to solve for y; we're trying to solve for x. We need to go back to our substitution and figure out what values of x correspond to these values of y.

It's important to remember that these values of y are just intermediate solutions. They're like stepping stones on our path to finding x. We've made progress, but we're not quite at the finish line yet. This is a common strategy in problem-solving: break a complex problem into smaller, more manageable steps.

Factoring is a powerful technique for solving quadratic equations, but it's not always easy to see the factors right away. If you're struggling with factoring, don't worry! The quadratic formula is always an option. It might involve a little more calculation, but it will always give you the solutions, even when factoring is difficult or impossible.

So, we've successfully solved for y, and we have two potential values. Now it's time to bring x back into the picture and see what these values of y tell us about the solutions to our original equation. Let's move on to the next step and substitute back to find x.

Step 4: Substitute Back and Solve for x

Now comes the fun part – bringing x back into the equation! Remember our substitution:

y = (2x + 6)

We found two values for y: 6 and 4. So, we need to solve two separate equations:

Case 1: y = 6

Substitute y = 6 into our substitution equation:

6 = 2x + 6

Subtract 6 from both sides:

0 = 2x

Divide both sides by 2:

x = 0

So, one solution is x = 0.

Case 2: y = 4

Substitute y = 4 into our substitution equation:

4 = 2x + 6

Subtract 6 from both sides:

-2 = 2x

Divide both sides by 2:

x = -1

So, our second solution is x = -1.

We've done it! We've successfully solved for x. We found two solutions: x = 0 and x = -1. These are the values of x that make our original equation true.

Substituting back is a crucial step in solving equations using substitution. It's where we connect our intermediate solutions (the values of y) back to the variable we were originally trying to solve for (x). It's like translating from one language back into another. We used the language of y to make the problem easier to solve, and now we're translating back into the language of x to get our final answer.

It's always a good idea to check our solutions by plugging them back into the original equation. This helps us catch any errors we might have made along the way. Let's do that in the next step.

Step 5: Check the Solutions

To make sure our solutions are correct, let's plug them back into the original equation:

(2x + 6)^2 - 10(2x + 6) + 22 = -2

Check x = 0

Substitute x = 0 into the equation:

(2(0) + 6)^2 - 10(2(0) + 6) + 22 = -2

(6)^2 - 10(6) + 22 = -2

36 - 60 + 22 = -2

-2 = -2

This is true, so x = 0 is a valid solution.

Check x = -1

Substitute x = -1 into the equation:

(2(-1) + 6)^2 - 10(2(-1) + 6) + 22 = -2

(4)^2 - 10(4) + 22 = -2

16 - 40 + 22 = -2

-2 = -2

This is also true, so x = -1 is a valid solution.

Both solutions check out! This gives us confidence that we've solved the equation correctly.

Checking our solutions is like the final proofread of a piece of writing. It's our last chance to catch any mistakes and make sure our answer is accurate. It's a step that's often overlooked, but it's a really important part of the problem-solving process. It not only confirms that our solutions are correct, but it also helps us solidify our understanding of the concepts involved.

So, by checking our solutions, we've not only verified our answer, but we've also reinforced our understanding of the equation and the steps we took to solve it.

Final Answer

The solutions to the equation (2x + 6)^2 - 10(2x + 6) + 22 = -2 are:

x = 0 and x = -1

We did it! We successfully solved a quadratic equation in quadratic form. Remember the key steps:

1. Simplify and rearrange the equation.
2. Make a substitution to create a standard quadratic equation.
3. Solve the quadratic equation for the new variable.
4. Substitute back to find the values of the original variable.
5. Check your solutions.

By following these steps, you can tackle any equation in quadratic form. Keep practicing, and you'll become a pro at solving these types of problems. Great job, guys! You've conquered a challenging equation today.