Solving 4t^2 - 14t + 7 = 0 Real Solutions In Simplest Form
Hey guys! Ever stumbled upon a quadratic equation that just seems daunting? Well, today, we're going to break down a classic example step by step. We'll tackle the equation 4t² - 14t + 7 = 0, finding all those real solutions in their simplest forms. Let’s dive in and make quadratic equations feel less like a puzzle and more like a piece of cake!
Understanding Quadratic Equations
First off, what exactly is a quadratic equation? In simple terms, it’s a polynomial equation of the second degree. That means it has a term with the variable raised to the power of 2 (like our 4t²), and no higher powers. The general form looks like this: ax² + bx + c = 0, where a, b, and c are constants, and 'a' isn't zero (otherwise, it wouldn’t be a quadratic equation anymore!).
Why are these equations so important? You'll find them popping up all over the place in math and real-world applications. Physics, engineering, even economics – quadratic equations are fundamental for modeling parabolic trajectories, optimizing areas, and a whole lot more. Mastering them is seriously a key skill.
Now, let's look at our specific equation: 4t² - 14t + 7 = 0. Here, 't' is our variable, and we need to find the values of 't' that make this equation true. Think of it like solving a mystery – we're on the hunt for the hidden values of 't'!
There are a few main methods we can use to crack this code. We could try factoring, which is like finding two expressions that multiply together to give us our quadratic equation. Sometimes, though, factoring can be tricky or even impossible with simple numbers. Another approach is completing the square, which involves rearranging the equation into a perfect square trinomial. But today, we're going to focus on the most reliable method: the quadratic formula.
The Mighty Quadratic Formula
The quadratic formula is our trusty tool for solving any quadratic equation, no matter how messy it looks. It's like the Swiss Army knife of algebra! The formula is:
t = (-b ± √(b² - 4ac)) / (2a)
Don't let it scare you – it's actually pretty straightforward once you get the hang of it. The plus-minus (±) symbol just means we'll have two possible solutions, one with addition and one with subtraction.
To use the formula, we need to identify the values of 'a', 'b', and 'c' from our equation 4t² - 14t + 7 = 0. Can you spot them? Here's the breakdown:
- a = 4 (the coefficient of the t² term)
- b = -14 (the coefficient of the t term – don't forget the negative sign!)
- c = 7 (the constant term)
Now that we've got our players, let's plug them into the formula and see what happens. This is where the magic begins!
Applying the Quadratic Formula Step-by-Step
Okay, let’s get our hands dirty and substitute those values into the quadratic formula. Remember, we have:
t = (-b ± √(b² - 4ac)) / (2a)
And our values are:
- a = 4
- b = -14
- c = 7
First, let's replace 'a', 'b', and 'c' in the formula:
t = (-(-14) ± √((-14)² - 4 * 4 * 7)) / (2 * 4)
See? We just swapped the letters for the numbers. Now, let's simplify this expression step by step. The first part is -(-14), which becomes simply 14 because two negatives make a positive. So, our equation now looks like this:
t = (14 ± √((-14)² - 4 * 4 * 7)) / (2 * 4)
Next, we need to tackle what's inside the square root. Let's calculate (-14)², which is 196. Then, we calculate 4 * 4 * 7, which equals 112. Now we have:
t = (14 ± √(196 - 112)) / (2 * 4)
Subtract 112 from 196, and we get 84. So, under the square root, we now have 84:
t = (14 ± √84) / (2 * 4)
Don't forget the denominator! 2 * 4 is simply 8. So, now our equation looks like this:
t = (14 ± √84) / 8
We’re getting closer! But before we call it a day, we need to simplify that square root. This is like tidying up our answer to make it look its best.
Simplifying the Radical
That square root of 84 might look a bit intimidating, but we can simplify it by finding perfect square factors. Think of it like breaking down a big number into smaller, more manageable pieces. We need to find factors of 84 where one of them is a perfect square (like 4, 9, 16, 25, etc.).
Can you spot a perfect square factor of 84? That's right, 4 is a factor! In fact, 84 can be written as 4 * 21. So, we can rewrite our square root like this:
√84 = √(4 * 21)
Now, here's a cool trick: the square root of a product is the product of the square roots. That means:
√(4 * 21) = √4 * √21
The square root of 4 is simply 2, so we have:
√84 = 2√21
See how much cleaner that looks? We've taken a messy square root and simplified it into something much easier to handle. Now, let's plug this back into our equation:
t = (14 ± 2√21) / 8
We’re almost there! We just have one more step to simplify this expression and get our final answer.
Final Simplification and Solutions
We've got our equation down to t = (14 ± 2√21) / 8, but we can make it even simpler. Notice that all the terms – 14, 2√21, and 8 – have a common factor of 2. This means we can divide everything by 2 to clean things up.
Dividing 14 by 2 gives us 7. Dividing 2√21 by 2 gives us just √21. And dividing 8 by 2 gives us 4. So, our equation simplifies to:
t = (7 ± √21) / 4
And there you have it! We've found our solutions. Because of the ± sign, we actually have two solutions:
- t₁ = (7 + √21) / 4
- t₂ = (7 - √21) / 4
These are the two real solutions to our equation 4t² - 14t + 7 = 0, expressed in their simplest form. We did it!
Wrapping Up
Solving quadratic equations can seem tricky at first, but with the right tools and a step-by-step approach, it becomes much more manageable. We’ve walked through using the quadratic formula, simplifying radicals, and finding the final solutions. Remember, the key is to break down the problem into smaller steps and tackle each one at a time.
So, next time you encounter a quadratic equation, don't sweat it! Just remember the quadratic formula, simplify those radicals, and you'll be solving equations like a pro in no time. Keep practicing, and you'll be amazed at how confident you become with these types of problems. You got this!