Solving X² = 7x + 4: A Quadratic Equation Guide

Hey guys! Today, we're diving into the exciting world of quadratic equations. Specifically, we're tackling the equation x_² = 7_x + 4 and figuring out which of the given options is the correct solution. Don't worry, it's not as scary as it looks! We'll break it down step by step, so you can confidently solve these problems yourself.

Understanding Quadratic Equations

Before we jump into solving this particular equation, let's quickly recap what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That basically means the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is _ax_² + bx + c = 0, where a, b, and c are constants (numbers), and a is not equal to 0. Recognizing this form is the first step in solving quadratic equations, as it allows us to apply standard methods like factoring, completing the square, or using the quadratic formula.

Our given equation, x_² = 7_x + 4, isn't quite in this standard form yet. To make it look like the standard form, we need to move all the terms to one side of the equation, leaving zero on the other side. This involves subtracting 7x and 4 from both sides of the equation. This rearrangement is crucial because the standard form allows us to easily identify the coefficients a, b, and c, which are essential for using the quadratic formula and other solution methods. Transforming the equation into standard form is not just about aesthetics; it's a necessary step that sets the stage for applying the appropriate mathematical tools to find the solutions.

Now, let’s consider why we can’t just guess the solutions or try random numbers. While that might work for very simple equations, quadratic equations often have solutions that are irrational numbers or complex numbers. These solutions can't be easily guessed, and that’s where the power of methods like the quadratic formula comes in. By putting the equation into the standard form, we’re setting ourselves up to use a reliable, universally applicable method for finding the solutions, no matter how complex they might be. So, remember, getting to that _ax_² + bx + c = 0 form is our crucial first step!

Getting the Equation into Standard Form

Okay, let’s get our hands dirty and transform our equation, x_² = 7_x + 4, into that standard form we just talked about. Remember, standard form is _ax_² + bx + c = 0. To achieve this, we need to move everything to the left-hand side of the equation. This means we need to get rid of the 7x and the 4 on the right-hand side. The way we do this, of course, is by performing the same operations on both sides of the equation to maintain balance.

So, the first thing we'll do is subtract 7x from both sides. This gives us x_² - 7_x = 4. See how we're getting closer? Now we just need to get rid of that 4 on the right-hand side. To do that, we'll subtract 4 from both sides. This gives us x_² - 7_x - 4 = 0. Bingo! We’ve successfully transformed our equation into standard form. Now, let's identify the coefficients a, b, and c. This is super important for the next step, which involves using the quadratic formula.

In our equation, x_² - 7_x - 4 = 0, the coefficient of the x² term (a) is 1 (because there's an implied 1 in front of the x²). The coefficient of the x term (b) is -7 (don't forget the negative sign!). And the constant term (c) is -4. These values are the keys to unlocking the solutions using the quadratic formula. So, make sure you're comfortable identifying a, b, and c from a quadratic equation in standard form. It's a fundamental skill for solving these kinds of problems, and it will make your life a whole lot easier as we move forward. Think of it as the secret code to the quadratic equation puzzle!

Applying the Quadratic Formula

Alright, now for the main event: using the quadratic formula. This formula is your best friend when it comes to solving quadratic equations, especially when they don't factor easily (and trust me, this one doesn't!). The quadratic formula is a magical tool that gives you the solutions for x in any quadratic equation in standard form. So, if you haven't already, make sure you memorize this bad boy. It's a lifesaver, seriously.

The formula itself looks a little intimidating at first, but it's really just a matter of plugging in the right numbers. Here it is:

x = (-b ± √(b_² - 4_ac)) / (2_a_)

Whoa, that's a mouthful, right? But don't worry, we'll break it down. Remember those coefficients a, b, and c we identified earlier? That's what we're going to plug into this formula. So, let's revisit our equation: x_² - 7_x - 4 = 0. We know that a = 1, b = -7, and c = -4. Now, it's just a matter of substituting these values into the formula and doing the math.

Carefully substitute each value, paying close attention to signs. This is a common area for mistakes, so double-check your work! Once you've substituted the values, you'll have an expression that you can simplify step-by-step. The first step is usually to calculate the value inside the square root (b_² - 4_ac). This part of the formula is called the discriminant, and it tells you a lot about the nature of the solutions. We'll talk more about the discriminant later, but for now, let's just focus on calculating it.

After you've calculated the discriminant, you can take the square root (if it's a perfect square) or leave it in radical form (if it's not). Remember, the ± sign in the formula means you'll actually have two solutions: one where you add the square root and one where you subtract it. This is because quadratic equations can have up to two distinct solutions. Finally, you'll simplify the entire expression to get your solutions for x. It might seem like a lot of steps, but with practice, it becomes second nature. And the satisfaction of finding those solutions? Totally worth it!

Plugging in the Values

Alright, guys, let's get practical and plug our values into the quadratic formula. This is where the magic happens! Remember our formula: x = (-b ± √(b_² - 4_ac)) / (2_a_). And remember our coefficients: a = 1, b = -7, and c = -4. Now, let’s carefully substitute these values into the formula. This step is crucial, so let's take our time and make sure we get it right.

First, we'll replace b with -7. Notice that there's a negative sign in front of the b in the formula, so we'll have -(-7), which becomes positive 7. Next, we'll replace a with 1 and c with -4. Now, our formula looks like this:

x = (7 ± √((-7)² - 4 * 1 * -4)) / (2 * 1)

See how we've carefully substituted each value? Now, we need to simplify this expression. The first thing we usually do is simplify the part under the square root, which is called the discriminant. So, let's calculate (-7)² - 4 * 1 * -4. Remember the order of operations (PEMDAS/BODMAS): we do exponents first, then multiplication, then addition and subtraction.

(-7)² is 49 (a negative number squared is positive). Then, 4 * 1 * -4 is -16. So, we have 49 - (-16), which is the same as 49 + 16, which equals 65. So, the expression under the square root simplifies to 65. Our formula now looks like this:

x = (7 ± √65) / 2

We're getting so close! Notice that 65 is not a perfect square, so we can't simplify the square root any further. That's okay! We just leave it as √65. Now, remember the ± sign? That means we actually have two solutions: one where we add √65 and one where we subtract it. This is a key characteristic of quadratic equations – they often have two solutions, reflecting the parabola's two potential intersections with the x-axis. So, let's separate those into our two final solutions.

Finding the Solutions

Okay, we're at the final stretch! We've plugged the values into the quadratic formula, simplified the expression, and now we're ready to write out our two solutions. Remember, the ± sign in the formula x = (7 ± √65) / 2 means we have two possibilities: one with addition and one with subtraction. This is because quadratic equations, which represent parabolas, often intersect the x-axis at two points, corresponding to these two solutions.

So, let's write out our first solution, where we add the square root:

_x_₁ = (7 + √65) / 2

And our second solution, where we subtract the square root:

_x_₂ = (7 - √65) / 2

These are our two solutions! They look a bit intimidating with the square root, but that's perfectly normal. Quadratic equations often have solutions that are irrational numbers, meaning they can't be expressed as simple fractions. This is why the square root remains in our answer. Now, let's compare these solutions to the options given in the problem. We have:

A. (-7 - √65) / 2, (-7 + √65) / 2 B. -7, 0 C. (7 - √65) / 2, (7 + √65) / 2 D. 7, 0

Looking at our solutions, _x_₁ = (7 + √65) / 2 and _x_₂ = (7 - √65) / 2, we can clearly see that they match option C. So, option C is the correct answer! We did it! We successfully solved the quadratic equation using the quadratic formula. Give yourself a pat on the back – you've conquered a challenging problem. This process might seem complex at first, but with practice, you'll become a quadratic equation-solving pro. Remember, the key is to break it down step by step, carefully substitute the values, and take your time with the arithmetic. You got this!

Conclusion

So, to wrap things up, the solutions to the quadratic equation x_² = 7_x + 4 are (7 - √65) / 2 and (7 + √65) / 2, which corresponds to option C. We tackled this problem by first putting the equation into standard form, then applying the mighty quadratic formula. We saw how crucial it is to correctly identify the coefficients a, b, and c, and how careful substitution into the formula is key to avoiding errors. We also learned that quadratic equations often have two solutions, which can be irrational numbers, and that's perfectly okay! The quadratic formula is our trusty tool for finding these solutions, no matter how complex they might look.

Solving quadratic equations is a fundamental skill in algebra, and it's used in many different areas of math and science. From physics to engineering to computer science, quadratic equations pop up all over the place. So, mastering this skill is a great investment in your mathematical future. Remember, practice makes perfect! The more you work with quadratic equations, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're part of the learning journey. And most importantly, have fun with it! Math can be challenging, but it can also be incredibly rewarding when you finally crack a tough problem. So, keep practicing, keep exploring, and keep those quadratic equations coming! You've got the tools, you've got the knowledge, now go out there and solve them! You're awesome, guys!