Solve X² - 12x + 40 = 0 Using The Quadratic Formula

by Andrew McMorgan 52 views

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics to tackle a super common problem: finding the exact solutions to a quadratic equation. Specifically, we'll be working with x² - 12x + 40 = 0. Now, I know sometimes these equations can look a little intimidating, especially when they involve imaginary numbers, but trust me, with the quadratic formula, it's totally doable and actually pretty cool once you get the hang of it. This formula is like your magic wand for solving any quadratic equation of the form ax² + bx + c = 0. It guarantees you'll find the solutions, whether they're real, complex, or even just repeated. So, grab your thinking caps, and let's break down how to use this powerful tool to nail down the exact solutions for our equation. We're aiming for precision here, so sticking to the formula is key. Get ready to see how this mathematical marvel works its magic!

Understanding the Quadratic Formula: Your Go-To Tool

Alright team, let's chat about the star of the show: the quadratic formula. This is the rockstar equation that helps us solve any quadratic equation, and our equation today, x² - 12x + 40 = 0, is no exception. Remember, a standard quadratic equation is always written in the form ax² + bx + c = 0. Before we plug in any numbers, it's crucial to identify what 'a', 'b', and 'c' are in our specific equation. In x² - 12x + 40 = 0, we can see that:

  • a = 1 (because there's an invisible '1' coefficient in front of x²)
  • b = -12 (don't forget the negative sign!)
  • c = 40

Got it? Awesome! Now, the quadratic formula itself is:

x = [-b ± √(b² - 4ac)] / 2a

This formula might look a bit long, but each part plays a vital role. The '±' symbol is super important because it tells us there are potentially two solutions – one where we add the square root part, and one where we subtract it. The term inside the square root, b² - 4ac, is called the discriminant. The discriminant is like a mini-fortune teller for our solutions. If it's positive, we get two different real solutions. If it's zero, we get one real solution (or two identical ones). But if it's negative, like it will be in our case, that's where the cool imaginary numbers come into play, giving us complex solutions.

Now, let's get down to business and plug our values for 'a', 'b', and 'c' into the formula. This is where the real fun begins, and we'll see exactly how these numbers unlock the secrets of our equation. It’s all about careful substitution and simplifying. We want the exact solutions, so we’ll keep things precise and avoid rounding until the very end, if even necessary. The goal is to get that final, clean answer that perfectly satisfies the original equation. So, no pressure, just good old-fashioned mathematical problem-solving. We’ve got this!

Plugging in the Values: Step-by-Step Calculation

Okay guys, we've identified our 'a', 'b', and 'c', and we have the quadratic formula ready to go. Now it's time for the main event: plugging those numbers in and crunching them down. This is where we see the magic happen and move closer to finding those exact solutions for x² - 12x + 40 = 0. Remember, a = 1, b = -12, and c = 40. Let's substitute these into the formula: x = [-b ± √(b² - 4ac)] / 2a.

First, let's handle the -b part. Since 'b' is -12, -b becomes -(-12), which simplifies to +12. Easy peasy!

Next up is the dreaded b² - 4ac under the square root. This is our discriminant, and it's where things might get a little interesting. Let's calculate it:

  • b² = (-12)² = 144
  • 4ac = 4 * (1) * (40) = 160

So, b² - 4ac = 144 - 160 = -16.

Uh oh! We have a negative number under the square root! This means our solutions will involve imaginary numbers. Don't panic; this is exactly what the quadratic formula is designed to handle. Remember that the square root of -1 is denoted by 'i'. So, √(-16) can be rewritten as √(16 * -1), which is √16 * √(-1). Since √16 is 4 and √(-1) is 'i', √(-16) = 4i.

Finally, let's deal with the denominator, 2a. Since 'a' is 1, 2a = 2 * 1 = 2.

Now, let's put all these pieces back into our quadratic formula:

x = [12 ± 4i] / 2

We're so close, guys! Just one more step to simplify this expression and find our exact solutions. It's all about distributing that division by 2 to both parts of the numerator. Keep your eyes peeled for the final answer!

Simplifying to Find the Exact Solutions

We've done the heavy lifting, team! We've plugged in our values for 'a', 'b', and 'c' into the quadratic formula and simplified the components. We arrived at x = [12 ± 4i] / 2. Now, for the grand finale: simplifying this expression to get our exact solutions. This step is all about making the answer as clean and concise as possible. Remember, we want the precise values that satisfy the equation x² - 12x + 40 = 0.

To simplify [12 ± 4i] / 2, we just need to divide each term in the numerator by the denominator, which is 2. Let's break it down:

  • Divide the real part: 12 / 2 = 6
  • Divide the imaginary part: 4i / 2 = 2i

So, when we combine these simplified parts, our solutions become:

x = 6 ± 2i

This means we have two exact solutions:

  1. x₁ = 6 + 2i
  2. x₂ = 6 - 2i

These are the precise, complex solutions to our quadratic equation. Notice how the '±' symbol beautifully splits into two distinct answers, each with a real part (6) and an imaginary part (±2i). This is the power of the quadratic formula – it doesn't shy away from complex numbers; it embraces them to give us the complete picture.

Now, let's quickly look at the options provided:

A. 8, 4 B. 6 ± 2i C. -6 ± 2i D. 12 ± 4i

Comparing our calculated solutions, 6 ± 2i, with the given options, it's clear that Option B is our correct answer! We totally nailed it, guys!

This process demonstrates that even when faced with a negative discriminant, leading to complex solutions, the quadratic formula is our reliable guide. It systematically leads us through the calculation, ensuring accuracy every step of the way. So, next time you see a quadratic equation, especially one that looks tricky, remember the quadratic formula. It’s your best friend for finding those exact solutions, no matter how complex they might seem at first glance. Keep practicing, and you'll be a quadratic formula pro in no time! Stay curious and keep solving!