Solving $4x^2 - 3x + 9 = 2x + 1$ Using The Quadratic Formula

Hey guys! Let's dive into solving a quadratic equation using the quadratic formula. We've got a fun problem on our hands: 4x² - 3x + 9 = 2x + 1. Our mission? Find the values of x that make this equation true. Buckle up, because we're about to break it down step by step.

Understanding the Quadratic Formula

Before we jump into solving, let's quickly revisit the quadratic formula. This formula is our trusty tool for solving any quadratic equation that's in the standard form ax² + bx + c = 0. The formula itself looks like this:

x = [-b ± √(b² - 4ac)] / 2a

Where a, b, and c are the coefficients from our quadratic equation. The ± symbol means we'll actually get two solutions: one using the plus sign and one using the minus sign. This accounts for the two possible roots of a quadratic equation.

Preparing the Equation

Our first task is to get our equation, 4x² - 3x + 9 = 2x + 1, into the standard form ax² + bx + c = 0. This means we need to move all the terms to one side of the equation, leaving zero on the other side.

Let's start by subtracting 2x from both sides:

4x² - 3x + 9 - 2x = 2x + 1 - 2x

This simplifies to:

4x² - 5x + 9 = 1

Now, subtract 1 from both sides to get zero on the right:

4x² - 5x + 9 - 1 = 1 - 1

Which gives us our equation in standard form:

4x² - 5x + 8 = 0

Great! Now we can clearly see that:

• a = 4
• b = -5
• c = 8

Plugging into the Formula

Now comes the exciting part – plugging our values of a, b, and c into the quadratic formula. Let's take it slow and steady to avoid any mistakes.

Remember the formula:

x = [-b ± √(b² - 4ac)] / 2a

Substitute the values:

x = [-(-5) ± √((-5)² - 4 * 4 * 8)] / (2 * 4)

Simplifying the Expression

Alright, let's simplify this step-by-step. First, let's deal with the negative signs and the exponent:

x = [5 ± √(25 - 4 * 4 * 8)] / 8

Next, let's multiply inside the square root:

x = [5 ± √(25 - 128)] / 8

Now, subtract inside the square root:

x = [5 ± √(-103)] / 8

Uh oh! We have a negative number inside the square root. This means our solutions will be complex numbers, involving the imaginary unit i, where i = √(-1).

Dealing with the Imaginary Unit

Let's rewrite √(-103) using i:

√(-103) = √(103 * -1) = √(103) * √(-1) = √103 * i

So our expression becomes:

x = [5 ± √103 * i] / 8

The Final Solutions

We've reached our final solutions! We can rewrite this as two separate solutions:

x = (5 + √103 * i) / 8

and

x = (5 - √103 * i) / 8

We can also write this in a more compact form:

x = (5 ± √103 * i) / 8

So, the values of x that satisfy the equation 4x² - 3x + 9 = 2x + 1 are complex numbers: (5 ± √103 * i) / 8. Comparing our solutions to the options given, we see that the correct answer is C. (5 ± √103 i) / 8.

Key Takeaways for Solving Quadratic Equations

Guys, solving quadratic equations might seem daunting at first, but with a clear process, it becomes much more manageable. Here are the key takeaways from our journey today, presented in a way that's super helpful for anyone tackling these problems:

• Standard Form is Your Friend: The quadratic formula works its magic only when your equation is in the standard form ax² + bx + c = 0. Always rearrange your equation to this form first. It's like making sure you have the right ingredients before you start cooking!
• Identify a, b, and c: Once you're in standard form, carefully identify the coefficients a, b, and c. These are the keys to unlocking the quadratic formula. A slight mistake here can throw off your entire solution, so double-check!
• The Quadratic Formula: Your Trusty Tool: Memorize the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. It's the bread and butter of solving these equations. Think of it as your superhero cape in the world of quadratic equations.
• Plug and Chug, Carefully: Substitute the values of a, b, and c into the formula. This is where precision counts. Take it one step at a time, especially with those negative signs and exponents. Imagine you're building a delicate watch – each piece needs to be placed perfectly.
• Simplify, Simplify, Simplify: After substituting, simplify the expression step by step. Follow the order of operations (PEMDAS/BODMAS) to avoid errors. It's like untangling a knot – patience and a systematic approach are key.
• Watch Out for the Discriminant: The expression b² - 4ac (inside the square root) is called the discriminant. It tells you about the nature of your solutions:
  • If b² - 4ac is positive, you have two distinct real solutions.
  • If b² - 4ac is zero, you have one real solution (a repeated root).
  • If b² - 4ac is negative, you have two complex solutions (involving i). This is like a weather forecast for your solutions – it gives you a sneak peek of what to expect.
• Embrace Complex Solutions: Don't freak out if you encounter a negative number inside the square root. This just means your solutions are complex numbers. Remember that √(-1) = i, and you're good to go. Think of i as a friendly visitor from the imaginary world, here to help you solve the problem.
• Final Answer Check: Once you've found your solutions, take a moment to check them against the options provided (if it's a multiple-choice question) or by plugging them back into the original equation. This is your safety net, ensuring you've arrived at the correct answer.
• Practice Makes Perfect: Like any skill, solving quadratic equations gets easier with practice. The more you do it, the more comfortable you'll become with the process. It's like riding a bike – the first few times might be wobbly, but soon you'll be cruising along smoothly.

By keeping these takeaways in mind, you'll be well-equipped to tackle any quadratic equation that comes your way. Remember, math isn't about magic; it's about understanding the process and applying it confidently.

Understanding the Discriminant: Nature of Roots

The discriminant, the b² - 4ac part under the square root in the quadratic formula, is super important because it tells us about the nature of the roots (or solutions) of our quadratic equation. Let's break it down, guys:

1. Positive Discriminant (b² - 4ac > 0): Two Distinct Real Roots

   When the discriminant is positive, it means we're taking the square root of a positive number. This gives us two different real numbers as solutions. Graphically, this means the parabola representing the quadratic equation intersects the x-axis at two distinct points. Think of it as the parabola having two