Solving The Quadratic Equation: -1/2 N^2 + 18 = 0
Hey guys! Ever stumbled upon a quadratic equation that looks a bit intimidating? Don't worry, we've all been there. Today, we're going to break down the equation -1/2 n^2 + 18 = 0 step-by-step, making sure you understand exactly how to find the solutions. So, grab your metaphorical pencils and let's dive in!
Understanding the Equation
Before we jump into solving, let's understand what we're dealing with. The equation -1/2 n^2 + 18 = 0 is a quadratic equation. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'n') is 2. Quadratic equations have a general form of ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants.
In our equation, we can identify the constants as follows:
- a = -1/2
- b = 0 (because there is no 'n' term)
- c = 18
The goal is to find the values of 'n' that make the equation true. These values are called the solutions or roots of the equation. Quadratic equations can have two, one, or no real solutions. In this case, we expect to find two solutions because of the nature of the equation. So, are you ready to uncover the secrets of this equation? Let’s start with isolating the n² term.
Isolating the n² Term
Our first step is to isolate the term containing 'n^2'. This means we want to get '-1/2 n^2' by itself on one side of the equation. To do this, we need to get rid of the '+ 18'. The inverse operation of addition is subtraction, so we'll subtract 18 from both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other to maintain the balance.
So, we have:
-1/2 n^2 + 18 - 18 = 0 - 18
This simplifies to:
-1/2 n^2 = -18
Great! We've taken the first step. Now that we have the term with n² isolated, we need to get rid of that pesky -1/2 coefficient. Think about what operation is happening between -1/2 and n². It's multiplication, right? So, to undo this multiplication, we need to do the inverse operation, which is division. But dividing by a fraction can be a bit tricky, so instead, we'll multiply by the reciprocal.
Eliminating the Coefficient
To eliminate the coefficient -1/2, we'll multiply both sides of the equation by its reciprocal. The reciprocal of -1/2 is -2/1, which is simply -2. So, we'll multiply both sides by -2. This is a crucial step, so let’s make sure we get it right. Multiplying by the reciprocal will effectively cancel out the fraction, leaving us with just the n² term.
Let's do it:
-2 * (-1/2 n^2) = -2 * (-18)
On the left side, -2 multiplied by -1/2 equals 1, so we're left with:
n^2 = -2 * (-18)
On the right side, -2 multiplied by -18 equals 36. Remember, a negative times a negative is a positive. So, our equation now looks like this:
n^2 = 36
Awesome! We're getting closer. We've managed to isolate n², which means we're just one step away from finding our solutions. Now, what operation will help us get 'n' by itself? You guessed it – we need to think about the inverse of squaring a number.
Finding the Square Root
Now we have n^2 = 36. To find the value of 'n', we need to take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots. This is because both a positive number and its negative counterpart, when squared, will result in a positive number. This is a key point in solving quadratic equations, so don't forget it!
So, let's take the square root of both sides:
√(n^2) = ±√36
The square root of n^2 is simply 'n', and the square root of 36 is 6. But remember, we need to consider both positive and negative roots, so we get:
n = ±6
This means that n can be either +6 or -6. These are the two solutions to our quadratic equation. We've done it! But before we celebrate, let's quickly check our answers to make sure they are correct.
Checking the Solutions
To make sure our solutions are correct, we'll substitute each value of 'n' back into the original equation and see if it holds true. This is a great habit to get into, as it helps prevent errors and ensures you have the correct answers. Let's start with n = 6.
Checking n = 6:
Substitute n = 6 into the original equation: -1/2 n^2 + 18 = 0
-1/2 * (6)^2 + 18 = 0
-1/2 * 36 + 18 = 0
-18 + 18 = 0
0 = 0
The equation holds true for n = 6. Now, let's check n = -6.
Checking n = -6:
Substitute n = -6 into the original equation: -1/2 n^2 + 18 = 0
-1/2 * (-6)^2 + 18 = 0
-1/2 * 36 + 18 = 0
-18 + 18 = 0
0 = 0
The equation also holds true for n = -6. So, we've verified that both 6 and -6 are indeed the solutions to the equation.
Conclusion
So, the solutions to the equation -1/2 n^2 + 18 = 0 are n = ±6. We found these solutions by isolating the n² term, eliminating the coefficient, and then taking the square root of both sides. Remember, always consider both positive and negative roots when solving quadratic equations. And don't forget to check your solutions to ensure accuracy! You guys nailed it! Keep practicing, and you'll become quadratic equation solving pros in no time!
