Solving Quadratic Equations A Step-by-Step Guide

Hey guys! Let's dive into solving a quadratic equation problem together. We're going to break down the equation (-(-11) ± √(625)) / (2 * 9) step by step so you can see exactly how it's done. This is super useful for anyone tackling algebra or just wanting to brush up on their math skills. So, grab your calculators, and let's get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, and a is not equal to zero. These equations pop up everywhere in math and science, from calculating the trajectory of a ball to designing bridges. Solving them is a fundamental skill, and the equation we're tackling today, while not in the standard form, uses similar principles.

The solutions to a quadratic equation are also known as its roots or zeros. These are the values of the variable that make the equation true. There are several ways to find these solutions, including factoring, completing the square, and using the quadratic formula. Our equation today is already partially solved, guiding us through a direct calculation of the roots using a simplified form of the quadratic formula.

The quadratic formula itself is a powerful tool derived from the method of completing the square, and it can solve any quadratic equation, regardless of whether it can be factored easily. The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

See how pieces of that formula show up in our starting equation? We're essentially looking at the tail end of a quadratic formula solution here, which makes it a great example for understanding the arithmetic involved in finding roots.

Breaking Down the Equation: (-(-11) ± √(625)) / (2 * 9)

Now, let's get our hands dirty with the actual equation: (-(-11) ± √(625)) / (2 * 9). This might look intimidating at first, but don't worry, we'll take it one piece at a time. The structure of the equation gives us a clear path to follow, and it's like a roadmap to the solutions of a quadratic equation.

The equation is set up to directly calculate the roots, which is awesome. It shows a point where the quadratic formula has already been applied, and we're down to the final arithmetic. This is a common scenario you'll encounter when working through problems, so it’s super valuable to understand each component.

The ± symbol is your first clue that we're dealing with a quadratic equation that likely has two solutions. This symbol means we need to perform the calculation twice: once with addition and once with subtraction. This is because quadratic equations can have up to two distinct real roots.

Step 1: Simplify the Numerator

Let's start by simplifying the numerator, which is (-(-11) ± √(625)). This part has two main components:

1. -(-11): This is a double negative, which simply becomes positive 11. So, -(-11) = 11.
2. √(625): This is the square root of 625. If you know your squares, you'll recognize that 25 * 25 = 625. So, √(625) = 25.

Now we can rewrite the numerator as:

11 ± 25

This means we have two possibilities: 11 + 25 and 11 - 25. We'll deal with these separately as we move forward. Remember, the ± symbol is the key to unlocking both solutions of the quadratic equation.

Step 2: Simplify the Denominator

The denominator is much simpler: 2 * 9. This is a straightforward multiplication:

2 * 9 = 18

So, our denominator is simply 18. Keeping the denominator clear and simple helps prevent errors and keeps the focus on the more complex numerator calculations.

Step 3: Calculate the Two Possible Solutions

Now we combine our simplified numerator and denominator. We have two cases to consider:

Case 1: Using Addition (11 + 25)

First, let's use the addition part of the ± symbol:

(11 + 25) / 18

Calculate the sum in the numerator:

36 / 18

Now, divide:

36 / 18 = 2

So, one solution is 2. Make sure to keep track of this first root, as it's one of the answers to our quadratic equation.

Case 2: Using Subtraction (11 - 25)

Next, we use the subtraction part of the ± symbol:

(11 - 25) / 18

Calculate the difference in the numerator:

-14 / 18

Now, divide. We can also simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

-14 / 18 = -7 / 9

So, our second solution is -7/9. This gives us the second root of the quadratic equation, completing our solution.

Final Solutions

Alright, we've done the heavy lifting and arrived at our final answers. The two solutions to the equation (-(-11) ± √(625)) / (2 * 9) are:

1. 2
2. -7/9

These are the two roots of the quadratic equation that this expression represents. If we were to graph the corresponding quadratic function, these values would be the points where the parabola intersects the x-axis.

Reviewing the Steps

Let's quickly recap the steps we took to solve this:

1. Simplified the numerator: We handled the double negative and the square root separately, then combined them using the ± symbol.
2. Simplified the denominator: This was a straightforward multiplication.
3. Calculated two solutions: We used both the addition and subtraction possibilities from the ± symbol to find our two roots.

Breaking the problem down into smaller, manageable steps makes it much easier to tackle. This approach is super useful for all sorts of math problems, so keep it in mind as you keep learning!

Real-World Applications of Quadratic Equations

You might be wondering,