Solving For X: Sum Of Values In A Quadratic Equation

Hey guys! Let's dive into a fun math problem today. We're going to figure out how to find the sum of all the x values that make a specific equation true. This involves a bit of algebra, but don't worry, we'll break it down step by step. So, grab your thinking caps, and let's get started!

Understanding the Equation

So, our main task here is to determine the sum of all values of x that make the given equation true. The equation we're dealing with is:

$9x^2 - 3x + 1 + 9x^2 - 3x = 20 - 10(3x^2 - 3x)$

This might look a little intimidating at first, but it's just a quadratic equation in disguise. Quadratic equations are those with the highest power of x being 2 (like $x^2$). To solve this, we'll need to simplify it and rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. Once we have it in this form, we can use some cool tricks to find the sum of the roots (that's just a fancy way of saying the values of x that solve the equation).

Before we start crunching numbers, let's take a moment to think about the big picture. We know we need to simplify and rearrange. We also know that the standard quadratic form is our goal. So, with that in mind, let's jump into the simplification process.

Step-by-Step Solution

Alright, let's break down this equation step by step. Our goal is to simplify it and get it into that standard quadratic form we talked about earlier.

1. Simplify the Equation

First, let's combine like terms on the left side of the equation:

$9x^2 - 3x + 1 + 9x^2 - 3x = 20 - 10(3x^2 - 3x)$

Combine the $x^2$ terms and the x terms:

$(9x^2 + 9x^2) + (-3x - 3x) + 1 = 20 - 10(3x^2 - 3x)$

This simplifies to:

$18x^2 - 6x + 1 = 20 - 10(3x^2 - 3x)$

2. Distribute on the Right Side

Now, let's distribute the -10 on the right side of the equation:

$18x^2 - 6x + 1 = 20 - 30x^2 + 30x$

3. Move All Terms to One Side

To get our equation into the standard quadratic form ($ax^2 + bx + c = 0$), we need to move all the terms to one side. Let's move everything to the left side:

$18x^2 - 6x + 1 + 30x^2 - 30x - 20 = 0$

4. Combine Like Terms Again

Now, let's combine those like terms again:

$(18x^2 + 30x^2) + (-6x - 30x) + (1 - 20) = 0$

This gives us:

$48x^2 - 36x - 19 = 0$

Okay, we've successfully transformed our original equation into a standard quadratic equation! Now, we're ready to use a cool trick to find the sum of the roots.

Finding the Sum of the Roots

Now comes the fun part! We've got our quadratic equation in the standard form: $48x^2 - 36x - 19 = 0$. Remember, we're not trying to find the individual values of x just yet; we're after the sum of those values.

Here's a neat trick: For any quadratic equation in the form $ax^2 + bx + c = 0$, the sum of the roots is given by the formula:

Sum of roots = $-\frac{b}{a}$

This is a super handy shortcut! No need to factor or use the quadratic formula (at least, not for this specific question).

1. Identify a and b

In our equation, $48x^2 - 36x - 19 = 0$:

• a is the coefficient of $x^2$, which is 48.
• b is the coefficient of x, which is -36.

2. Apply the Formula

Now, let's plug these values into our formula:

Sum of roots = $-\frac{b}{a} = -\frac{-36}{48}$

3. Simplify

Simplify the fraction:

Sum of roots = $\frac{36}{48}$

Both 36 and 48 are divisible by 12, so let's reduce the fraction:

Sum of roots = $\frac{3}{4}$

And there you have it! The sum of all the x values that satisfy the equation is $\frac{3}{4}$. Isn't that a cool trick?

Why This Works: A Glimpse into Quadratic Theory

Okay, so we used this nifty formula (Sum of roots = $-\frac{b}{a}$) to find the answer, but you might be wondering, "Why does that work?" Let's take a peek behind the curtain and see some of the theory that makes this trick tick.

The Quadratic Formula and Roots

First, let's quickly revisit the quadratic formula. This formula is your trusty tool for finding the roots (solutions) of any quadratic equation in the form $ax^2 + bx + c = 0$:

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

Notice the "±" symbol? This means there are actually two possible solutions (roots) for a quadratic equation. Let's call them $x_1$ and $x_2$:

$x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$

$x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$

Summing the Roots

Now, what happens if we add these two roots together? Let's do it:

$x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}$

Notice that both fractions have the same denominator ($2a$), so we can combine them:

$x_1 + x_2 = \frac{(-b + \sqrt{b^2 - 4ac}) + (-b - \sqrt{b^2 - 4ac})}{2a}$

Look what happens! The $\sqrt{b^2 - 4ac}$ terms cancel each other out:

$x_1 + x_2 = \frac{-b - b}{2a}$

$x_1 + x_2 = \frac{-2b}{2a}$

And finally, we can simplify:

$x_1 + x_2 = -\frac{b}{a}$

Boom! We've derived the formula for the sum of the roots. This shows you why the formula works – it's a direct result of the structure of the quadratic formula itself.

The Bigger Picture

This little derivation highlights something really cool about math: formulas aren't just magic tricks. They're built on logical foundations and interconnected ideas. Understanding the "why" behind the formulas makes them much easier to remember and use.

So, next time you use the formula for the sum of the roots, you can remember that it's not just a random shortcut. It's a neat consequence of how quadratic equations and their solutions work!

Conclusion

So, guys, we've successfully tackled this problem! We took a seemingly complex equation, simplified it into a standard quadratic form, and then used a clever formula to find the sum of the roots. Remember, the key is to break down the problem into smaller, manageable steps. And don't be afraid to use those mathematical shortcuts – they're there to make our lives easier!

If you ever encounter a similar problem, remember the steps we took: simplify, rearrange, identify a and b, and apply the formula. You'll be solving quadratic equations like a pro in no time! Keep practicing, and math will become less of a challenge and more of a fun puzzle to solve.