Solving 2x² + 5x = 12 A Step-by-Step Guide
Hey everyone! Let's dive into solving a classic quadratic equation. We've got 2x² + 5x = 12, and we're going to break it down step by step. Quadratic equations might seem intimidating at first, but with a clear method, they become pretty manageable. So, grab your pencils, and let’s get started!
Understanding Quadratic Equations
Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') 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 zero. If 'a' were zero, the equation would become linear, not quadratic. The 'a' term is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term.
Quadratic equations pop up all over the place in mathematics and real-world applications. You'll find them in physics, engineering, economics, and even computer science. They're used to model parabolic trajectories (like the path of a ball thrown in the air), optimize areas and volumes, and solve problems involving rates of change. Understanding how to solve them is a fundamental skill in mathematics. Solving a quadratic equation means finding the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation. A quadratic equation can have up to two real solutions, one real solution (which is a repeated root), or two complex solutions. The nature of the solutions depends on the discriminant, which we'll touch on later.
There are several methods to solve quadratic equations, and we'll be using one of the most common: factoring. Other methods include using the quadratic formula, completing the square, and graphing. Each method has its advantages and is suitable for different types of quadratic equations. Factoring is often the quickest method when the quadratic expression can be easily factored. The quadratic formula is a universal method that works for all quadratic equations. Completing the square is a method that's useful for deriving the quadratic formula and solving certain types of equations. Graphing can give a visual representation of the solutions, which are the x-intercepts of the parabola. For our equation, factoring will be the most efficient route.
Step 1: Setting the Equation to Zero
The first crucial step in solving any quadratic equation is to set it equal to zero. This aligns our equation with the standard form ax² + bx + c = 0, which is essential for most solution methods. Our original equation is 2x² + 5x = 12. To set it to zero, we need to subtract 12 from both sides of the equation. This maintains the equality and moves all terms to one side. So, let's subtract 12 from both sides:
2x² + 5x - 12 = 12 - 12
This simplifies to:
2x² + 5x - 12 = 0
Now we have our quadratic equation in the standard form, with a = 2, b = 5, and c = -12. This form is necessary for applying factoring techniques or using the quadratic formula. It's a simple step, but it's a critical one. Without setting the equation to zero, we can't correctly identify the coefficients and constants, which are vital for finding the solutions. Think of it as preparing the ingredients before you start cooking – you need everything in the right place before you can create the final dish. This rearrangement allows us to see the equation in its most workable form, setting the stage for the next steps in our solution process.
Step 2: Factoring the Quadratic Expression
Now that our equation is in the form 2x² + 5x - 12 = 0, we can move on to the factoring stage. Factoring involves breaking down the quadratic expression into two binomial expressions (expressions with two terms) that, when multiplied together, give us the original quadratic expression. This might sound tricky, but with a bit of practice, it becomes a straightforward process. Our goal is to find two binomials (px + q) and (rx + s) such that:
(px + q)(rx + s) = 2x² + 5x - 12
To do this, we need to consider the coefficients and the constant term in our quadratic expression. We're looking for two numbers that multiply to give the product of the coefficient of x² (2) and the constant term (-12), which is 2 * -12 = -24, and add up to the coefficient of x (5). This is a common technique in factoring quadratic expressions. Let's list the factors of -24 and see which pair adds up to 5:
- -1 and 24 (sum = 23)
- -2 and 12 (sum = 10)
- -3 and 8 (sum = 5)
- -4 and 6 (sum = 2)
- -6 and 4 (sum = -2)
- -8 and 3 (sum = -5)
- -12 and 2 (sum = -10)
- -24 and 1 (sum = -23)
We found our pair! -3 and 8 multiply to -24 and add up to 5. Now, we'll use these numbers to rewrite the middle term (5x) in our quadratic expression. We split 5x into -3x + 8x:
2x² - 3x + 8x - 12 = 0
Next, we'll use a technique called factoring by grouping. We group the first two terms and the last two terms:
(2x² - 3x) + (8x - 12) = 0
Now, we factor out the greatest common factor (GCF) from each group. From the first group (2x² - 3x), the GCF is x:
x(2x - 3)
From the second group (8x - 12), the GCF is 4:
4(2x - 3)
Now, our equation looks like this:
x(2x - 3) + 4(2x - 3) = 0
Notice that we have a common binomial factor, (2x - 3), in both terms. We can factor this out:
(2x - 3)(x + 4) = 0
And there we have it! We've successfully factored the quadratic expression into two binomials. This factored form is crucial for finding the solutions to our equation. The next step involves using the zero-product property to determine the values of x that make the equation true.
Step 3: Applying the Zero-Product Property
We've reached a significant milestone in our solution process. Our equation is now factored into the form (2x - 3)(x + 4) = 0. This is where the zero-product property comes into play. The zero-product property is a fundamental concept in algebra that states: if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if we have A * B = 0, then either A = 0 or B = 0 (or both). This property is incredibly useful for solving equations that are factored.
Applying this property to our equation, (2x - 3)(x + 4) = 0, we set each factor equal to zero:
2x - 3 = 0 or x + 4 = 0
Now, we have two simple linear equations to solve. These are much easier to handle than the original quadratic equation. Let's solve each one separately. First, let's solve 2x - 3 = 0. To isolate 'x', we'll add 3 to both sides of the equation:
2x - 3 + 3 = 0 + 3
2x = 3
Next, we'll divide both sides by 2:
2x / 2 = 3 / 2
x = 3/2
So, one solution is x = 3/2. Now, let's solve the second equation, x + 4 = 0. To isolate 'x', we'll subtract 4 from both sides:
x + 4 - 4 = 0 - 4
x = -4
Therefore, our second solution is x = -4. We've found two values for x that satisfy our factored equation. These values are the roots or solutions of the original quadratic equation. The zero-product property is a powerful tool because it transforms a complex problem (solving a quadratic equation) into two simpler problems (solving linear equations). It allows us to break down the problem into manageable parts and find the solutions methodically. This step is crucial for finding the final answers and completing the solution process.
Step 4: Verifying the Solutions
We've arrived at what we believe are the solutions to our quadratic equation: x = 3/2 and x = -4. But before we declare victory, it's essential to verify these solutions. Verifying our solutions ensures that we haven't made any mistakes along the way and that our answers truly satisfy the original equation. This step is a crucial part of the problem-solving process in mathematics and helps us build confidence in our results. To verify our solutions, we'll substitute each value of x back into the original equation, 2x² + 5x = 12, and see if both sides of the equation are equal.
Let's start with x = 3/2. We'll substitute this value into the equation:
2(3/2)² + 5(3/2) = 12
First, we'll square 3/2:
2(9/4) + 5(3/2) = 12
Next, we'll multiply:
18/4 + 15/2 = 12
To add these fractions, we need a common denominator, which is 4. So, we'll convert 15/2 to 30/4:
18/4 + 30/4 = 12
Now, we can add the fractions:
48/4 = 12
Simplify the fraction:
12 = 12
Great! When we substitute x = 3/2 into the original equation, both sides are equal. This confirms that x = 3/2 is indeed a solution. Now, let's verify the second solution, x = -4. We'll substitute this value into the equation:
2(-4)² + 5(-4) = 12
First, we'll square -4:
2(16) + 5(-4) = 12
Next, we'll multiply:
32 - 20 = 12
Now, we'll subtract:
12 = 12
Excellent! When we substitute x = -4 into the original equation, both sides are also equal. This confirms that x = -4 is a solution as well. Since both values satisfy the original equation, we can confidently say that our solutions are correct. Verifying our solutions is like the final check in a recipe – it ensures that the dish (our solution) tastes just right. It's a valuable habit to develop in mathematics, as it helps prevent errors and reinforces our understanding of the problem-solving process.
Final Answer
Alright, we've reached the end of our journey! After meticulously setting up the equation, factoring it, applying the zero-product property, and verifying our solutions, we've successfully solved the quadratic equation 2x² + 5x = 12. Our solutions are:
x = 3/2
x = -4
These are the two values of x that make the equation true. We used factoring, which is a powerful technique for solving quadratic equations when the expression can be easily factored. Remember, the key steps were:
- Setting the equation to zero.
- Factoring the quadratic expression.
- Applying the zero-product property.
- Verifying the solutions.
Quadratic equations are fundamental in mathematics and have wide-ranging applications in various fields. Mastering the techniques to solve them is a valuable skill. Whether you're dealing with physics problems, engineering designs, or economic models, the ability to solve quadratic equations will undoubtedly come in handy. Keep practicing, and you'll become a pro at solving these equations in no time! Great job, everyone, on tackling this problem with us. Keep up the excellent work, and remember, mathematics is all about practice and perseverance. Until next time, happy solving!
