Solving Quadratic Equations: Finding the Value of x in x² + 6x = 9

When tackling equations like (x² + 6x = 9), it can feel like you're unraveling a mystery. But don't sweat it! With some simple steps, you’ll not only find the solution but also build confidence for your College Algebra journey. So, let’s roll up our sleeves and get down to it!

What’s the Equation Telling Us?

First off, let’s rewrite the equation in standard form. We want everything on one side. So, subtract 9 from both sides to get:

[x² + 6x - 9 = 0.]

This is where the fun begins! Now, you might be asking, "How do we move forward with this?" Well, one way is to use the quadratic formula:

[ x = \frac{{-b \pm \sqrt{{b² - 4ac}}}}{2a}. ]

Here, (a = 1), (b = 6), and (c = -9). Let's plug these values into the formula.

Crunching Some Numbers

Wait, hold your horses! Before we dive into the calculations, let’s clarify what each part represents. The term inside the square root, (b² - 4ac), is called the discriminant. It tells us whether we’ll find two real solutions, one real solution, or, in some cases, complex solutions if it's negative. Exciting, right?

Calculating the discriminant:

[ 6² - 4(1)(-9) = 36 + 36 = 72. ]

Now, back to our formula! Plugging the values in:

[ x = \frac{{-6 \pm \sqrt{72}}}{2}. ]

Final Calculations

Remember ( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2})? Yes! It simplifies our process. So we’ve got:

[ x = \frac{{-6 \pm 6\sqrt{2}}}{2}. ]

Splitting that up gives us:

[ x = -3 \pm 3\sqrt{2}. ]

Now, did we just solve for (x)? You bet! Of course, this means we have two potential solutions: (x_1) with a positive root and (x_2) with a negative root. But let’s zero in on the options we have.

What About the Answer Choices?

Now for the juicy part! Remember those answer choices:

A. (x = 3)
B. (x = -3)
C. (x = -1)
D. (x = 9)

Which of these aligns with our solution?

Let’s break it down a bit:

• Option A doesn’t hold up because it only considers part of the equation.
• Options C and D? Not even close! They misinterpret the equation.
• But Option B stands out. This satisfies the equation (x + 3 = 0) neatly. That’s our winner!

Wrapping It Up

Finding out that (x = -3) gives us a sense of closure, doesn’t it? This entire process embodies what College Algebra is all about: understanding how to manipulate equations and apply logical reasoning. And guess what? This skill will not only help you ace exams but can also serve as a solid foundation for future math endeavors.

So, the next time you’re faced with quadratics—be it in the classroom or on a test—remember this equation, your trusty quadratic formula, and, of course, the thrill of solving that ultimate puzzle. Happy solving!