Solving Quadratic Equation: 6x^2 + 6 = 12x + 18

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Hey guys! Today, we're diving into the world of quadratic equations and tackling a specific problem. We need to find the solutions for the equation 6x^2 + 6 = 12x + 18. Don't worry, it might look intimidating at first, but we'll break it down step by step so it's super easy to follow. We'll explore how to manipulate the equation, get it into a standard form, and then use our trusty tools to find those elusive solutions. Think of this as a puzzle – we've got all the pieces, and now we just need to put them together in the right way.

Understanding Quadratic Equations

First things first, let's chat about what a quadratic equation actually is. A quadratic equation is basically a polynomial equation of the second degree. What does that mean? Well, it means the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation is usually written as ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. If a were zero, then the x^2 term would disappear, and we'd be left with a linear equation instead. Quadratic equations pop up all over the place in math and science, from calculating trajectories in physics to modeling growth and decay in biology. They're a fundamental concept, so understanding them is really important. Remember the standard form – it's the key to unlocking many solution methods. Recognizing the a, b, and c values will be crucial when we move on to using the quadratic formula or completing the square, two common techniques for finding solutions. We also call the solutions to a quadratic equation the roots or zeros of the equation. This is because these are the x-values that make the equation equal to zero. Keep these terms in mind – they'll be helpful as we proceed.

Step-by-Step Solution

Okay, let's get our hands dirty and actually solve the equation 6x^2 + 6 = 12x + 18. The first thing we need to do is rearrange the equation into that standard form we talked about earlier: ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation, leaving zero on the other side. Let's start by subtracting 12x from both sides: 6x^2 - 12x + 6 = 18. Now, we need to get rid of that 18 on the right side, so we subtract 18 from both sides: 6x^2 - 12x - 12 = 0. Great! Now our equation is in standard form. Notice that all the coefficients (the numbers in front of the x terms and the constant term) are divisible by 6. This means we can simplify the equation by dividing both sides by 6: x^2 - 2x - 2 = 0. This simplified equation is much easier to work with. Now that we have our simplified quadratic equation, we need to decide how to solve it. We could try factoring, but this equation doesn't factor easily. So, we'll turn to the quadratic formula, a powerful tool that always works for solving quadratic equations. Get ready to put those memorization skills to the test, or just keep the formula handy! The quadratic formula is given by: x = [-b ± √(b^2 - 4ac)] / 2a. Remember those a, b, and c values from the standard form? Now they come into play. In our equation, x^2 - 2x - 2 = 0, we have a = 1, b = -2, and c = -2.

Applying the Quadratic Formula

Alright, we've got our quadratic formula ready and our a, b, and c values identified. Now it's time to plug those values into the formula and see what happens. Remember, the quadratic formula is: x = [-b ± √(b^2 - 4ac)] / 2a. We have a = 1, b = -2, and c = -2. Let's substitute these values in: x = [-(-2) ± √((-2)^2 - 4 * 1 * -2)] / (2 * 1). Okay, that looks a bit messy, but we'll simplify it step by step. First, let's deal with the negative signs: x = [2 ± √(4 + 8)] / 2. Now, let's simplify the expression under the square root: x = [2 ± √12] / 2. We can simplify the square root of 12 by factoring out the largest perfect square. 12 is equal to 4 times 3, and the square root of 4 is 2, so we can write √12 as 2√3. Our equation now looks like this: x = [2 ± 2√3] / 2. Notice that both terms in the numerator have a common factor of 2. We can divide both terms by 2 to simplify further: x = 1 ± √3. And there we have it! We've found the solutions to the quadratic equation. The solutions are x = 1 + √3 and x = 1 - √3. These are the two values of x that will make the original equation true. High five yourself – you've just conquered a quadratic equation!

Verifying the Solutions

Now that we've found our solutions, it's always a good idea to check our work and make sure they're correct. We can do this by plugging each solution back into the original equation 6x^2 + 6 = 12x + 18 and seeing if it holds true. Let's start with the solution x = 1 + √3. Substituting this value into the equation, we get: 6(1 + √3)^2 + 6 = 12(1 + √3) + 18. This looks like a bit of a beast, but we'll tackle it carefully. First, we need to expand (1 + √3)^2. Remember that (a + b)^2 = a^2 + 2ab + b^2, so (1 + √3)^2 = 1 + 2√3 + 3 = 4 + 2√3. Now we can substitute this back into our equation: 6(4 + 2√3) + 6 = 12(1 + √3) + 18. Next, we distribute the 6 on the left side and the 12 on the right side: 24 + 12√3 + 6 = 12 + 12√3 + 18. Simplifying both sides, we get: 30 + 12√3 = 30 + 12√3. Woohoo! The equation holds true for x = 1 + √3. Now let's check the other solution, x = 1 - √3. Substituting this value into the equation, we get: 6(1 - √3)^2 + 6 = 12(1 - √3) + 18. Again, we need to expand (1 - √3)^2. Using the formula (a - b)^2 = a^2 - 2ab + b^2, we get (1 - √3)^2 = 1 - 2√3 + 3 = 4 - 2√3. Substituting this back into our equation: 6(4 - 2√3) + 6 = 12(1 - √3) + 18. Distributing the 6 on the left side and the 12 on the right side: 24 - 12√3 + 6 = 12 - 12√3 + 18. Simplifying both sides, we get: 30 - 12√3 = 30 - 12√3. Awesome! The equation also holds true for x = 1 - √3. Since both solutions check out, we can be confident that we've solved the equation correctly.

Conclusion

So, there you have it! We successfully found the solutions to the quadratic equation 6x^2 + 6 = 12x + 18. We took the following steps to solve this equation:

  1. Rearranged the equation into standard form (ax^2 + bx + c = 0).
  2. Simplified the equation by dividing by a common factor.
  3. Applied the quadratic formula (x = [-b ± √(b^2 - 4ac)] / 2a).
  4. Simplified the resulting expression to find the solutions.
  5. Verified the solutions by plugging them back into the original equation.

Our solutions are x = 1 + √3 and x = 1 - √3. Remember, practice makes perfect, so keep working on those quadratic equations, and you'll become a pro in no time. Until next time, happy solving!