Solve X² - 7 = X + 5: Step-by-Step Guide

Hey guys! Ever stumbled upon a quadratic equation that looks like a puzzle? Don't worry, it happens to the best of us! Today, we're going to break down the quadratic equation x² - 7 = x + 5 into easy-to-understand steps. We'll transform it from a seemingly complex problem into something totally manageable. Quadratic equations might seem intimidating at first, but with a little bit of algebra magic, you'll be solving them like a pro in no time. So, let’s dive in and unlock the secrets of this equation together!

What are Quadratic Equations?

Before we get started on our specific equation, let's quickly recap what quadratic equations actually are. In simple terms, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Understanding this basic structure is crucial because it provides a framework for solving these equations. When we look at our equation, x² - 7 = x + 5, we can already see the x² term, which tells us it's a quadratic equation. Now, the key is to rearrange this equation into the standard form so we can apply our solving techniques. This involves moving all terms to one side, leaving zero on the other, which helps us identify the 'a', 'b', and 'c' coefficients. These coefficients are super important because they're the building blocks for using methods like factoring, completing the square, or the quadratic formula. By understanding the anatomy of a quadratic equation, we’re setting ourselves up for success in solving it. So, remember the standard form – it’s your guide in the quadratic world!

Step 1: Rearranging the Equation

The first step to solving any quadratic equation, including our x² - 7 = x + 5, is to bring it 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. Let's start by subtracting 'x' from both sides of the equation. This gives us x² - x - 7 = 5. Notice how we're keeping the equation balanced by doing the same operation on both sides. This is a fundamental principle in algebra – whatever you do to one side, you must do to the other to maintain equality. Now, we need to get rid of the '+5' on the right side. To do this, we subtract 5 from both sides. This gives us our rearranged equation: x² - x - 12 = 0. Ta-da! We've successfully transformed our original equation into the standard form. Now, can you see the 'a', 'b', and 'c' values? Here, 'a' is the coefficient of x² (which is 1), 'b' is the coefficient of 'x' (which is -1), and 'c' is the constant term (which is -12). Identifying these coefficients is like finding the key ingredients for our solution recipe. Once we have them, we can move on to the next step and choose the best method to solve the equation.

Step 2: Choosing a Solution Method

Now that we have our equation in the standard form, x² - x - 12 = 0, we need to decide on the best method to solve it. There are several options available, including factoring, completing the square, and the quadratic formula. Each method has its strengths, and the best choice often depends on the specific equation. For this particular equation, factoring looks like a promising approach. Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that multiply together to give the original expression. It's a quick and efficient method when it works, and it relies on our ability to recognize patterns in the coefficients. Another method, completing the square, is a bit more involved but is a powerful technique that can solve any quadratic equation. It involves manipulating the equation to create a perfect square trinomial. The quadratic formula is the most general method and can be used to solve any quadratic equation, regardless of whether it can be factored or not. It's a bit like the Swiss Army knife of quadratic equation solving! However, it can be a bit more computationally intensive than factoring. So, for x² - x - 12 = 0, let's try factoring first. It's often the fastest route if we can find the right factors. If factoring doesn't work out, we can always fall back on the quadratic formula. Choosing the right method is like picking the right tool for the job – it makes the task much easier!

Step 3: Factoring the Quadratic Equation

Okay, let's give factoring a shot! We have our equation x² - x - 12 = 0. To factor this, we need to find two numbers that multiply to give -12 (the 'c' term) and add up to give -1 (the 'b' term). Think of it like a little number puzzle. What two numbers fit the bill? After a bit of thought, we can see that the numbers -4 and 3 work perfectly. Why? Because (-4) * (3) = -12 and (-4) + (3) = -1. Awesome! Now that we've found our numbers, we can rewrite the quadratic expression as a product of two binomials: (x - 4)(x + 3) = 0. See how the -4 and +3 slot into the binomials? Factoring is like unlocking a secret code – once you find the right combination, the solution is within reach. Now, we're not quite done yet. We've factored the equation, but we still need to find the values of 'x' that make the equation true. This is where the zero-product property comes in, which we'll use in the next step. But for now, let's appreciate the power of factoring – it's a neat way to simplify a quadratic equation and get us closer to the solution.

Step 4: Applying the Zero-Product Property

Now that we've factored our equation into (x - 4)(x + 3) = 0, we can use a cool trick called the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. Makes sense, right? If you multiply two numbers and get zero, one of them has to be zero. So, in our case, either (x - 4) must equal zero, or (x + 3) must equal zero. This gives us two simple equations to solve: x - 4 = 0 and x + 3 = 0. Let's solve the first one: x - 4 = 0. To isolate 'x', we add 4 to both sides, giving us x = 4. Great! We've found one solution. Now, let's tackle the second equation: x + 3 = 0. To isolate 'x' here, we subtract 3 from both sides, which gives us x = -3. Fantastic! We've found our second solution. The zero-product property is like a magic key that unlocks the solutions once we've factored the quadratic equation. It transforms a single equation into two simpler ones, making the problem much easier to solve. So, remember this property – it's a powerful tool in your algebra arsenal!

Step 5: Verifying the Solutions

We've found two potential solutions for our equation: x = 4 and x = -3. But before we declare victory, it's always a good idea to verify our solutions. This means plugging each value of 'x' back into the original equation x² - 7 = x + 5 to make sure they actually work. Let's start with x = 4. Substituting this into the equation, we get (4)² - 7 = 4 + 5. Simplifying, we have 16 - 7 = 9, which further simplifies to 9 = 9. Hooray! This is a true statement, so x = 4 is indeed a solution. Now, let's check x = -3. Substituting this into the original equation, we get (-3)² - 7 = -3 + 5. Simplifying, we have 9 - 7 = 2, which simplifies to 2 = 2. Another true statement! So, x = -3 is also a solution. Verifying our solutions is like double-checking our work – it ensures we haven't made any mistakes along the way and gives us confidence in our answer. It's a crucial step in problem-solving, not just in algebra but in many areas of life. So, always take the time to verify your solutions – it's worth the extra effort!

Conclusion

Alright, guys! We did it! We successfully solved the quadratic equation x² - 7 = x + 5 step by step. We rearranged the equation into standard form, chose the factoring method, applied the zero-product property, and verified our solutions. Our solutions are x = 4 and x = -3. You've now seen how a seemingly complex equation can be tackled with a systematic approach and a few key techniques. Quadratic equations might seem daunting at first, but by breaking them down into manageable steps, you can conquer them with confidence. Keep practicing, and you'll become a quadratic equation-solving master in no time! Remember, math is like a puzzle – each piece fits together to create a beautiful solution. And you've just put another piece in your mathematical puzzle-solving skills. Awesome job!