Unlock The Solution To 2x^2 + 8x = X^2 - 162

by Andrew McMorgan 45 views

Hey math whizzes and equation explorers! Today, we're diving headfirst into a super interesting quadratic equation: 2x2+8x=x2−1622x^2 + 8x = x^2 - 162. If you're wondering, "What is the only solution of 2x2+8x=x2−1622x^2+8x=x^2-162?", you've come to the right place, guys. We're going to break this down step-by-step, making sure you understand every move. Quadratic equations can seem a bit daunting at first, but trust me, with a solid approach, they become totally manageable, and even, dare I say, fun! So, grab your calculators, your notebooks, and let's get ready to unravel this mathematical mystery. We'll be using some classic algebraic techniques to isolate the variable and find the value(s) of 'x' that make this equation true. It's all about rearranging, simplifying, and finally, uncovering those golden solutions.

Getting the Equation into Standard Form

The first, and arguably most crucial, step when tackling any quadratic equation is to get it into its standard form. For quadratic equations, this means getting everything onto one side, leaving zero on the other. The standard form looks like this: ax2+bx+c=0ax^2 + bx + c = 0. Why is this important? Because it allows us to use established methods like factoring, completing the square, or the quadratic formula to find our solutions. So, let's take our given equation, 2x2+8x=x2−1622x^2 + 8x = x^2 - 162, and start moving terms around. We want to gather all the x2x^2 terms, all the xx terms, and all the constant terms on one side. A good strategy here is to subtract x2x^2 from both sides. This will leave us with 2x2−x2+8x=−1622x^2 - x^2 + 8x = -162, which simplifies to x2+8x=−162x^2 + 8x = -162. Awesome! We're closer. Now, we need to get that −162-162 over to the left side. To do that, we simply add 162162 to both sides of the equation. So, x2+8x+162=−162+162x^2 + 8x + 162 = -162 + 162. This gives us our final standard form: x2+8x+162=0x^2 + 8x + 162 = 0. See? Not so scary after all. We've successfully transformed the original equation into a clean, usable format. This form is our gateway to finding the solution(s) for 'x'. Remember, the goal is always to simplify and organize. By putting it into ax2+bx+c=0ax^2 + bx + c = 0 format, we're setting ourselves up for success with the next steps. It's like preparing your ingredients before you start cooking – essential for a delicious outcome!

Applying the Quadratic Formula

Now that our equation is in the standard form, x2+8x+162=0x^2 + 8x + 162 = 0, we can identify the coefficients aa, bb, and cc. In this case, a=1a = 1 (the coefficient of x2x^2), b=8b = 8 (the coefficient of xx), and c=162c = 162 (the constant term). With these values in hand, we can now employ the mighty quadratic formula, which is our trusty tool for solving any equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The formula is: $x = rac{-b pm

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sqrt{b^2 - 4ac}}{2a}$. Let's plug in our values: $a=1$, $b=8$, and $c=162$. This gives us $x = rac{-8 pm // Another comment here, will be removed. $sqrt{8^2 - 4(1)(162)}}{2(1)}$. Now, we need to simplify the expression under the square root, which is called the discriminant. The discriminant, $b^2 - 4ac$, tells us a lot about the nature of our solutions. Let's calculate it: $8^2 - 4(1)(162) = 64 - 648$. Uh oh, we have $64 - 648 = -584$. So, the expression under the square root is $-584$. This means our equation involves the square root of a negative number. When the discriminant is negative, it indicates that there are no *real* solutions. Instead, the solutions are complex or imaginary. The problem asks for "the only solution," which often implies a real number solution in typical algebra contexts unless otherwise specified. Since we have a negative discriminant, let's pause and consider if we made any mistakes or if the question implies a specific type of solution. In many standard high school or introductory college algebra problems, if a quadratic equation yields a negative discriminant, the answer is stated as "no real solutions." However, if we are expected to work with complex numbers, we can proceed. Let's assume for a moment the question might be leading us to explore complex solutions. If so, we'd write $\sqrt{-584}$ as $\sqrt{584}i$. We can simplify $\sqrt{584}$ by finding its prime factorization. $584 = 2 imes 292 = 2 imes 2 imes 146 = 2 imes 2 imes 2 imes 73$. So, $\sqrt{584} = \sqrt{4 imes 146} = 2\sqrt{146}$. Thus, $\sqrt{-584} = 2\sqrt{146}i$. Now, let's plug this back into the quadratic formula: $x = rac{-8 pm // Another comment, to be removed. $2\sqrt{146}i}{2}$. Simplifying this by dividing both terms in the numerator by 2, we get $x = -4 pm // Final comment, to be removed. $\sqrt{146}i$. This gives us two complex solutions: $x = -4 + \sqrt{146}i$ and $x = -4 - \sqrt{146}i$. However, the original question asks for "the *only* solution." This phrasing strongly suggests there should be a single, unique answer, typically a real one in this type of problem. The fact that we arrived at complex solutions, and two of them at that, makes us re-evaluate. Let's double-check our algebra. $2x^2 + 8x = x^2 - 162$. Subtract $x^2$: $x^2 + 8x = -162$. Add 162: $x^2 + 8x + 162 = 0$. The coefficients are $a=1$, $b=8$, $c=162$. The discriminant is $b^2 - 4ac = 8^2 - 4(1)(162) = 64 - 648 = -584$. Our calculations seem correct. When a quadratic equation has a negative discriminant, it means the parabola it represents never touches the x-axis. Therefore, it has no real roots. If the question *strictly* implies a single real solution, then there isn't one. However, if the question is perhaps phrased a bit loosely and is just asking for *a* solution, or *all* solutions, then the complex ones are it. But the word "only" is very specific. This could mean one of two things: either there's a typo in the original equation, or the question is trying to trick us, or it's a poorly phrased question that expects us to say there are no real solutions, and thus, no *single* real solution. Given the phrasing, and the result, the most accurate answer in the realm of real numbers is that there is *no* solution. ### Interpreting the