Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of quadratic equations. If you've ever felt lost trying to solve these equations, don't worry! This guide will break it down step by step, making it super easy to understand. We'll tackle eight different equations together, so you'll have plenty of practice. Let's get started!

What are Quadratic Equations?

First things first, let's define what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x, y, m, or t in our examples) is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation are called its roots. We're going to explore different methods to find these roots, including factoring and using the quadratic formula. Understanding the structure of these equations is crucial for solving them efficiently. So, keep this general form in mind as we move forward.

Why are Quadratic Equations Important?

You might be wondering, why bother learning about quadratic equations? Well, they show up in a ton of real-world applications! From physics and engineering to economics and computer science, quadratic equations are used to model various phenomena. For example, they can describe the trajectory of a projectile, the shape of a satellite dish, or even the optimal pricing strategy for a product. Mastering quadratic equations opens doors to solving complex problems in many different fields. Think of it as adding a powerful tool to your problem-solving toolkit!

Methods for Solving Quadratic Equations

There are several ways to solve quadratic equations, but we'll focus on two primary methods today: factoring and the quadratic formula. Each method has its strengths and weaknesses, and knowing when to use each one can save you a lot of time and effort. Let's take a closer look at each method.

1. Factoring

Factoring is a method that involves breaking down the quadratic expression into a product of two binomials. This method is particularly useful when the equation can be easily factored. The basic idea behind factoring is to reverse the process of expanding two binomials. For example, if we have (x + 2)(x + 3) = 0, we can easily see that the roots are x = -2 and x = -3. However, not all quadratic equations can be factored easily, so it's essential to recognize when this method is most suitable.

To factor a quadratic equation in the form ax² + bx + c = 0, you need to find two numbers that multiply to c and add up to b. Once you find these numbers, you can rewrite the quadratic expression as a product of two binomials and then set each binomial equal to zero to find the roots. Factoring is often the quickest method when it works, but it requires some practice to become proficient.

2. Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. It works for any quadratic equation, regardless of whether it can be factored easily or not. The formula is derived from the method of completing the square and is given by:

x = (-b ± √(b² - 4ac)) / 2a

Where a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0. The quadratic formula might look intimidating at first, but it's a powerful tool once you get the hang of it. All you need to do is identify the coefficients a, b, and c, plug them into the formula, and simplify. The ± sign indicates that there are usually two solutions: one with the plus sign and one with the minus sign.

The part under the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it's zero, there is one real root (a repeated root). If it's negative, there are two complex roots. Understanding the discriminant can give you a quick insight into the solutions of the quadratic equation.

Solving the Equations

Alright, let's put these methods into action and solve the eight equations you provided. We'll go through each one step-by-step, so you can see exactly how it's done.

1) x² + 7x - 60 = 0

For this equation, we can use factoring. We need to find two numbers that multiply to -60 and add up to 7. Those numbers are 12 and -5. So, we can factor the equation as:

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

Setting each factor equal to zero gives us the roots:

x + 12 = 0 => x = -12

x - 5 = 0 => x = 5

So, the roots are x = -12 and x = 5. See how factoring can make things super quick?

2) y² + 10y - 24 = 0

Let's try factoring again. We need two numbers that multiply to -24 and add up to 10. These numbers are 12 and -2. Factoring the equation, we get:

(y + 12)(y - 2) = 0

Setting each factor to zero:

y + 12 = 0 => y = -12

y - 2 = 0 => y = 2

Therefore, the roots are y = -12 and y = 2. Nice and easy!

3) m² + m - 90 = 0

This one can also be factored. We need two numbers that multiply to -90 and add up to 1. The numbers are 10 and -9. So, the factored form is:

(m + 10)(m - 9) = 0

Setting the factors to zero:

m + 10 = 0 => m = -10

m - 9 = 0 => m = 9

Thus, the roots are m = -10 and m = 9. Factoring is really shining here!

4) 3t² + 7t + 4 = 0

This one looks a bit trickier to factor because of the coefficient 3 in front of the t² term. We can still factor it, but let's walk through the process carefully. We're looking for two binomials that multiply to give us the original quadratic. After some trial and error (or using techniques like the AC method), we find:

(3t + 4)(t + 1) = 0

Setting each factor to zero:

3t + 4 = 0 => 3t = -4 => t = -4/3

t + 1 = 0 => t = -1

So, the roots are t = -4/3 and t = -1. Factoring is still doable, but it's getting a bit more complex.

5) 3x² + 32x + 80 = 0

This equation looks challenging to factor directly. When factoring gets too complicated, the quadratic formula is our best friend! Let's identify a, b, and c:

a = 3, b = 32, c = 80

Now, plug these values into the quadratic formula:

x = (-32 ± √(32² - 4 * 3 * 80)) / (2 * 3)

Simplify:

x = (-32 ± √(1024 - 960)) / 6

x = (-32 ± √64) / 6

x = (-32 ± 8) / 6

Now, we have two possible solutions:

x = (-32 + 8) / 6 = -24 / 6 = -4

x = (-32 - 8) / 6 = -40 / 6 = -20/3

So, the roots are x = -4 and x = -20/3. See how the quadratic formula comes to the rescue when factoring is tough?

6) 2x² + 9x - 486 = 0

This equation also looks difficult to factor, so let's use the quadratic formula again. Identify a, b, and c:

a = 2, b = 9, c = -486

Plug the values into the quadratic formula:

x = (-9 ± √(9² - 4 * 2 * -486)) / (2 * 2)

Simplify:

x = (-9 ± √(81 + 3888)) / 4

x = (-9 ± √3969) / 4

x = (-9 ± 63) / 4

Two possible solutions:

x = (-9 + 63) / 4 = 54 / 4 = 27/2

x = (-9 - 63) / 4 = -72 / 4 = -18

Thus, the roots are x = 27/2 and x = -18. Quadratic formula for the win!

7) 3x² - 6x + 3 = 0

Before jumping to the quadratic formula, let's see if we can simplify this equation. Notice that all the coefficients are divisible by 3. Let's divide the entire equation by 3:

x² - 2x + 1 = 0

Now, this looks much easier to factor! We need two numbers that multiply to 1 and add up to -2. Those numbers are -1 and -1. So, the factored form is:

(x - 1)(x - 1) = 0

Or, simply:

(x - 1)² = 0

This gives us one repeated root:

x - 1 = 0 => x = 1

So, the root is x = 1 (with multiplicity 2). Always look for simplifications first!

8) 9x² + 6x + 1 = 0

This equation also looks factorable. It's a perfect square trinomial! Notice that 9x² is (3x)², 1 is 1², and 6x is 2 * (3x) * 1. So, we can factor it as:

(3x + 1)² = 0

Setting the factor to zero:

3x + 1 = 0 => 3x = -1 => x = -1/3

So, the root is x = -1/3 (with multiplicity 2). Recognizing patterns can save you time!

Conclusion

We've solved eight quadratic equations using both factoring and the quadratic formula. You've seen that factoring is often quicker when it works, but the quadratic formula is a reliable method that works every time. The key is to practice and become comfortable with both methods. Remember to always look for opportunities to simplify the equation before diving into solving it.

Keep practicing, and you'll become a quadratic equation-solving pro in no time! If you have any questions, feel free to ask. Happy solving! 🚀