Solving Quadratic Equations: Find The Roots

Hey guys! Ever found yourself staring at a quadratic equation and feeling totally lost? Don't sweat it! We're going to break down a simple one today, step by step, so you can tackle these problems like a pro. Let’s dive into solving the equation $-7y - y^2 = 0$. This is a classic quadratic equation, and we’ll explore a straightforward way to find its solutions. So, grab your pencils, and let’s get started!

Understanding the Equation

First, let's rewrite the equation in the standard quadratic form, which is $ay^2 + by + c = 0$. In our case, the equation $-7y - y^2 = 0$ can be rearranged as $-y^2 - 7y = 0$. To make it look even more familiar and to follow mathematical convention, we can multiply the entire equation by -1, which gives us $y^2 + 7y = 0$. See? It already looks a bit friendlier!

Now, let's identify the coefficients: here, $a = 1$, $b = 7$, and $c = 0$. Understanding these coefficients is crucial because they play a significant role in various methods of solving quadratic equations, such as using the quadratic formula or completing the square. However, for this particular equation, we’ll use a simpler and more direct method: factoring. Factoring is often the quickest way to solve a quadratic equation when it’s applicable, and in this case, it definitely is.

Why is factoring so cool? Well, it breaks down the quadratic expression into a product of simpler expressions. When we set each of these simpler expressions equal to zero, we can easily find the values of $y$ that satisfy the original equation. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental concept in algebra and makes factoring an incredibly powerful tool.

Factoring the Equation

Okay, now let’s factor the equation $y^2 + 7y = 0$. Notice that both terms have a common factor of $y$. We can factor out $y$ from both terms: $y(y + 7) = 0$. Factoring is like reverse distribution; we're pulling out the common element to simplify the expression. In this case, $y$ is the hero that makes our equation much easier to solve. When you see a common factor like this, always take advantage of it!

Now we have the equation in a factored form: $y(y + 7) = 0$. This form tells us that the product of $y$ and $(y + 7)$ is zero. According to the zero-product property, this means that either $y = 0$ or $y + 7 = 0$. This property is the cornerstone of solving equations by factoring, and it allows us to split our problem into two simpler equations.

The zero-product property states that if $ab = 0$, then either $a = 0$ or $b = 0$ (or both). In our case, $a$ is $y$ and $b$ is $(y + 7)$. By applying this property, we can easily find the solutions for $y$. This step is critical because it transforms a quadratic equation into two linear equations, which are much easier to solve. So, remember the zero-product property – it’s your best friend when it comes to solving factored equations!

Solving for y

Alright, let's solve each of these simpler equations. First, we have $y = 0$. This is already one of our solutions! It's straightforward and requires no further work. Sometimes, the simplest solutions are the easiest to overlook, but in this case, it's right there in front of us. So, our first solution is $y = 0$.

Next, we need to solve $y + 7 = 0$. To isolate $y$, we subtract 7 from both sides of the equation: $y + 7 - 7 = 0 - 7$, which simplifies to $y = -7$. This is our second solution. By performing this simple algebraic manipulation, we've found the other value of $y$ that satisfies the original equation.

So, our two solutions are $y = 0$ and $y = -7$. These are the values of $y$ that make the equation $-7y - y^2 = 0$ true. To double-check, you can plug these values back into the original equation to confirm that they satisfy it. This is always a good practice to ensure that your solutions are correct.

Verifying the Solutions

To verify our solutions, we'll substitute each value back into the original equation, $-7y - y^2 = 0$, and see if the equation holds true.

First, let's check $y = 0$:

$-7(0) - (0)^2 = 0 - 0 = 0$

So, $y = 0$ is indeed a solution.

Now, let's check $y = -7$:

$-7(-7) - (-7)^2 = 49 - 49 = 0$

Thus, $y = -7$ is also a valid solution. Both solutions satisfy the original equation, giving us confidence in our results.

Verifying solutions is an important step in solving any equation. It helps to catch any mistakes that might have been made during the solving process. By plugging the solutions back into the original equation, we can ensure that the solutions are accurate and that we have not made any algebraic errors. This practice is especially useful when dealing with more complex equations, where errors are more likely to occur.

Final Answer

Therefore, the solutions to the equation $-7y - y^2 = 0$ are $y = 0$ and $y = -7$. These are the only two values of $y$ that will make the equation true. You’ve successfully navigated through this quadratic equation! Great job! Knowing how to solve these types of equations is a fundamental skill in algebra, and it opens the door to solving more complex problems in mathematics and science.

So, remember the steps we took: rearrange the equation, factor out the common factor, apply the zero-product property, and solve for $y$. With practice, you’ll become more comfortable and confident in solving quadratic equations. Keep up the great work, and you’ll be acing those math problems in no time!

A. 0 and -7