Quadratic Formula: Solve X^2-7x+6=0

Hey mathletes! Today, we're diving deep into the magical world of quadratic equations and tackling one that's been thrown around: $x^2 - 7x + 6 = 0$. You've probably seen this bad boy pop up in your algebra classes, and maybe you've even sweated a bit trying to figure out what values of $x$ make it true. Well, fret not, because we're going to break it down using the Quadratic Formula, a trusty tool in any mathematician's belt. This formula is your best friend when factoring gets a little tricky, or when you just want a surefire way to find those roots. So, grab your calculators, maybe a snack, and let's get solving!

Understanding the Quadratic Formula

First things first, guys, what exactly is the Quadratic Formula? For any quadratic equation in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients and $a$ is not zero, the solutions for $x$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This formula looks a bit intimidating at first glance, I know. It's like a secret code for unlocking the secrets of parabolas and their intersections with the x-axis. But trust me, once you get the hang of plugging in the numbers, it becomes super straightforward. The part under the square root, $b^2 - 4ac$, is called the discriminant, and it tells us a lot about the nature of the roots (whether they're real, imaginary, or repeated). For our problem, $x^2 - 7x + 6 = 0$, we need to identify our $a$, $b$, and $c$ values. In this case, $a = 1$ (since there's an invisible 1 in front of $x^2$), $b = -7$ (don't forget the negative sign!), and $c = 6$. We're going to substitute these values into the formula and work our way through it step-by-step. It’s like following a recipe, but instead of making cookies, we’re finding the values that make our equation balance out perfectly. So, remember this formula, write it down, tattoo it on your brain (okay, maybe just write it down), because it’s going to save you a ton of time and effort.

Applying the Formula to Our Equation

Alright, team, let's get down to business with our specific equation: $x^2 - 7x + 6 = 0$. We’ve identified our coefficients: $a=1$, $b=-7$, and $c=6$. Now, we plug these numbers into the Quadratic Formula.

$$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(6)}}{2(1)}$$

See? We just replaced $a$, $b$, and $c$ with their respective values. The first step here is to simplify the terms within the formula. Let's start with the $b$ term: $-(-7)$ becomes $7$. Easy peasy. Next, let's tackle the part under the square root, the discriminant. $(-7)^2$ is $49$, and $4(1)(6)$ is $24$. So, inside the square root, we have $49 - 24$, which equals $21$. Our denominator is $2(1)$, which is just $2$. So now our formula looks like this:

$$x = \frac{7 \pm \sqrt{21}}{2}$$

Now, here's where things get interesting. The square root of $21$ isn't a nice, clean whole number. It's an irrational number. This means our solutions for $x$ will also be irrational. The formula gives us two possible solutions because of the $\pm$ (plus or minus) sign. One solution will use the plus sign, and the other will use the minus sign. This is totally normal and expected when using the quadratic formula. It accounts for the two points where a parabola can cross the x-axis.

Calculating the Two Solutions

So, we have our simplified formula: $x = \frac{7 \pm \sqrt{21}}{2}$. To find our two distinct solutions for $x$, we'll separate the $\pm$ into two distinct calculations. This is where we get our specific answers, guys.

Solution 1 (using the '+' sign):

$$x_1 = \frac{7 + \sqrt{21}}{2}$$

Solution 2 (using the '-' sign):

$$x_2 = \frac{7 - \sqrt{21}}{2}$$

These are the exact solutions. If you were asked for approximate decimal values, you'd use a calculator to find $\sqrt{21} \approx 4.58$. Then you could calculate:

$$x_1 \approx \frac{7 + 4.58}{2} = \frac{11.58}{2} \approx 5.79$$

$$x_2 \approx \frac{7 - 4.58}{2} = \frac{2.42}{2} \approx 1.21$$

However, the question asks for the exact solutions, and looking at the multiple-choice options provided (A. $x=5,6$; B. $x=2,3$; C. $x=1,6$; D. $x=-2,-3$; E. $x=-1,-6$), none of them match our derived irrational solutions. This suggests there might be a misunderstanding in the question's premise or the provided options, as the Quadratic Formula applied correctly to $x^2-7x+6=0$ yields irrational roots. Let's double-check if the equation was perhaps meant to be factorable, or if there's a typo. Often, when quadratic equations have integer solutions, they can be factored easily. For $x^2 - 7x + 6 = 0$, we're looking for two numbers that multiply to $6$ and add up to $-7$. Those numbers are $-1$ and $-6$. So, the factored form is $(x-1)(x-6) = 0$. Setting each factor to zero gives us $x-1=0 ightarrow x=1$ and $x-6=0 ightarrow x=6$. These are integer solutions! This means the original problem statement, while asking to use the Quadratic Formula, might have intended an equation that could be solved by factoring, or the options provided are for a different equation. If we were to use the quadratic formula on an equation that does have integer roots like $(x-1)(x-6)=0$, which expands to $x^2 - 7x + 6 = 0$, then the quadratic formula should yield $x=1$ and $x=6$. Let's re-evaluate our application of the formula. Ah, I see the confusion! My apologies, guys. I got a little sidetracked by the irrational result. Let's re-run the calculation carefully, focusing on the discriminant.

Re-evaluating the Discriminant and Roots

Let's go back to the heart of the Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For $x^2 - 7x + 6 = 0$, we have $a=1$, $b=-7$, and $c=6$. Plugging these in again:

$$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(6)}}{2(1)}$$

Simplifying:

$$x = \frac{7 \pm \sqrt{49 - 24}}{2}$$

$$x = \frac{7 \pm \sqrt{25}}{2}$$

Hold up! I made a calculation error in the previous step. The discriminant is $49 - 24 = 25$, not $21$. My bad! This changes everything, and it's a perfect example of why double-checking your work is so crucial, especially with numbers.

Now that we have $\sqrt{25}$, which is a nice, clean $5$, we can proceed:

$$x = \frac{7 \pm 5}{2}$$

This means we have two distinct solutions:

Solution 1 (using the '+' sign):

$$x_1 = \frac{7 + 5}{2} = \frac{12}{2} = 6$$

Solution 2 (using the '-' sign):

$$x_2 = \frac{7 - 5}{2} = \frac{2}{2} = 1$$

So, the solutions are $x=6$ and $x=1$. This is fantastic because these are whole numbers, which often appear in textbook problems and multiple-choice questions. It means our equation is indeed factorable, and the Quadratic Formula worked like a charm, just as it should!

Final Answer and Conclusion

After meticulously applying the Quadratic Formula to the equation $x^2 - 7x + 6 = 0$, we found our two solutions to be $x=6$ and $x=1$. Let's quickly check these against the options provided:

A. $x=5,6$ B. $x=2,3$ C. $x=1,6$ D. $x=-2,-3$ E. $x=-1,-6$

Our calculated solutions, $x=1$ and $x=6$, perfectly match option C. It's always a good feeling when your math work lines up with the choices, right? This whole process highlights the power and reliability of the Quadratic Formula. Even when an equation looks like it might have complex roots, the formula will always guide you to the correct answer. Remember, guys, it's not just about getting the answer; it's about understanding the process. The Quadratic Formula is a fundamental concept in algebra, and mastering it will open doors to solving even more complex mathematical challenges. Keep practicing, keep questioning, and don't be afraid to go back and check your steps – you never know when a small slip can lead you astray, like my little discriminant detour! Happy solving!