Solve: -7x-60=x^2+10x & Find Solutions

Hey guys! Today, we're going to tackle a classic algebra problem: completing an equivalent equation and finding the solutions for a quadratic equation. Specifically, we'll be working with the equation $-7x - 60 = x^2 + 10x$. Don't worry, we'll break it down step by step so it's super easy to follow.

Transforming the Equation: Setting the Stage for Solutions

Our initial equation is $-7x - 60 = x^2 + 10x$. To solve this, the first thing we need to do is rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form is crucial because it allows us to use methods like factoring or the quadratic formula to find the solutions. When dealing with quadratic equations, it's essential to get all terms on one side. This ensures we can effectively identify the coefficients a, b, and c, which are vital for further steps like factoring or applying the quadratic formula. Our main keyword here is understanding how to manipulate the equation into a solvable format. Think of it as prepping the ingredients before you start cooking – you need everything in the right place!

To get there, we'll add $7x$ and $60$ to both sides of the equation. This might seem like a simple step, but it's a fundamental algebraic manipulation that keeps the equation balanced. Remember, whatever you do to one side, you must do to the other. This gives us:

$x^2 + 10x + 7x + 60 = 0$

Now, we combine like terms. We have two 'x' terms, $10x$ and $7x$, which we can add together. This simplification is a key step in solving any algebraic equation. It helps to reduce the complexity and makes the equation easier to work with. Combining like terms is like organizing your tools before a project – it makes the whole process smoother and more efficient.

This gives us:

$x^2 + 17x + 60 = 0$

Great! We've now got our equation in the standard quadratic form. This is a major step forward. We've successfully transformed the initial equation into a form that's much easier to solve. This standard form is our launchpad for finding the solutions, the values of 'x' that make the equation true. Think of it as translating a sentence into a language you understand – now we can finally decode its meaning!

Factoring the Quadratic: Unlocking the Solutions

Now that we have our equation in the standard form ($x^2 + 17x + 60 = 0$), we can move on to factoring. Factoring is a technique that involves breaking down the quadratic expression into two binomials. This method works well when the quadratic expression can be easily factored, and it's often quicker than using the quadratic formula. When factoring, we are essentially reversing the process of expansion. We're looking for two expressions that, when multiplied together, give us the original quadratic. This is like finding the pieces of a puzzle that fit perfectly together to create the whole picture.

We need to find two numbers that add up to 17 (the coefficient of the 'x' term) and multiply to 60 (the constant term). This might sound tricky, but with a little thought and maybe some trial and error, it's quite manageable. Think of it as a detective game, where you're searching for the right clues to solve the mystery. Remember, the numbers need to satisfy both conditions – addition and multiplication – to be the correct factors.

Let's think about the factors of 60: 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, 6 and 10. Which of these pairs adds up to 17? Bingo! 5 and 12. This is a crucial step, and finding the right factors is key to solving the equation. It's like finding the right key to unlock a door – once you have it, you can move forward.

So, we can rewrite the quadratic equation as:

$(x + 5)(x + 12) = 0$

This is the factored form of our equation. We've successfully broken down the quadratic expression into two binomials. This factored form is super helpful because it directly leads us to the solutions of the equation. It's like having a treasure map that shows you exactly where to find the hidden gold – in this case, the solutions for 'x'!

Finding the Solutions: The Zero Product Property

We've now reached the final stage: finding the solutions for 'x'. We have the equation in factored form: $(x + 5)(x + 12) = 0$. This is where the Zero Product Property comes into play. The Zero Product Property is a fundamental principle in algebra that states that if the product of two factors is zero, then at least one of the factors must be zero. This property is a powerful tool for solving equations, especially those in factored form. It essentially transforms the problem of solving a single equation into solving two simpler equations.

In our case, we have two factors: $(x + 5)$ and $(x + 12)$. For their product to be zero, either $(x + 5)$ must be zero, or $(x + 12)$ must be zero (or both!). This understanding is crucial for applying the Zero Product Property effectively. It's like understanding the rules of a game – you need to know how the pieces move to play the game well.

So, we set each factor equal to zero:

$x + 5 = 0$ or $x + 12 = 0$

Now we solve each of these simple equations for 'x'. To solve $x + 5 = 0$, we subtract 5 from both sides. To solve $x + 12 = 0$, we subtract 12 from both sides. These are basic algebraic manipulations that isolate 'x' and give us the solutions. Think of it as untangling a knot – you're carefully undoing the connections to reveal the individual strands.

This gives us:

$x = -5$ or $x = -12$

These are the solutions to our quadratic equation! We've successfully found the values of 'x' that make the original equation true. This is like reaching the destination on a journey – all the steps we've taken have led us to this final result. These solutions are the roots of the equation, the points where the quadratic function crosses the x-axis.

Summarizing Our Journey: The Complete Solution

Let's recap what we've done. We started with the equation $-7x - 60 = x^2 + 10x$. We rearranged it into the standard quadratic form, factored the quadratic expression, and then used the Zero Product Property to find the solutions. This process is a common and effective way to solve quadratic equations. It's like following a recipe – each step is important, and when followed correctly, it leads to a successful outcome.

The equivalent equation in factored form is:

$(x + 5)(x + 12) = 0$

The solutions to the equation $-7x - 60 = x^2 + 10x$ are:

$x = -5$ and $x = -12$

So there you have it, guys! We've successfully completed the equation and found the solutions. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!