Solve $-(x-3)^2-3=7$ As A Complex Number

Let's dive into solving the equation $-(x-3)^2-3=7$ and express the answer as a complex number in the standard form $a ext{ extpm } b i$. This problem involves algebraic manipulation and understanding how to represent solutions that might extend beyond real numbers into the realm of complex numbers. We'll break down each step carefully, ensuring clarity and accuracy, so you can confidently tackle similar problems. Get ready to flex those mathematical muscles!

Isolating the Squared Term

The first crucial step in solving equations of this form is to isolate the term containing the square. Our equation is $-(x-3)^2-3=7$. To begin isolating $-(x-3)^2$, we need to get rid of the '-3' on the left side. We can do this by adding 3 to both sides of the equation. This operation maintains the equality of the equation. So, we have:

$-(x-3)^2 - 3 + 3 = 7 + 3$

This simplifies to:

$-(x-3)^2 = 10$

Now, we need to deal with the negative sign in front of the squared term. To isolate $(x-3)^2$, we multiply both sides of the equation by -1. Remember, multiplying both sides by the same non-zero number preserves the equality. Therefore:

$(-1) * -(x-3)^2 = (-1) * 10$

Which gives us:

$(x-3)^2 = -10$

At this stage, we have successfully isolated the squared term. This is a significant milestone because the next step involves taking the square root of both sides, which will introduce the possibility of complex numbers if the right-hand side is negative, as it is here.

Taking the Square Root

With the equation in the form $(x-3)^2 = -10$, we can now take the square root of both sides. When we take the square root of a squared term, we get the base term. However, it's essential to remember that when taking the square root of both sides of an equation, we must consider both the positive and negative roots. This is because both $(+ ext{number})^2$ and $(- ext{number})^2$ result in the same positive number. So, applying the square root to both sides yields:

$\sqrt{(x-3)^2} = \pm\sqrt{-10}$

This simplifies to:

$x-3 = \pm\sqrt{-10}$

Now, we need to address the square root of -10. The presence of the negative sign under the square root is where complex numbers come into play. We know that the imaginary unit, denoted by $i$, is defined as $\sqrt{-1}$. Therefore, we can rewrite $\sqrt{-10}$ as:

$\sqrt{-10} = \sqrt{10 * -1} = \sqrt{10} * \sqrt{-1} = \sqrt{10}i$

So, our equation becomes:

$x-3 = \pm \sqrt{10}i$

This shows that our solutions will involve the imaginary unit $i$, confirming that we are indeed working with complex numbers. The $\pm$ symbol indicates that there are two distinct solutions stemming from this step.

Solving for x

The final step in solving for $x$ is to isolate it completely. Currently, we have $x-3$ on the left side. To get $x$ by itself, we need to add 3 to both sides of the equation. Again, adding the same number to both sides maintains the balance of the equation.

$x - 3 + 3 = 3 \pm \sqrt{10}i$

This simplifies directly to:

$x = 3 \pm \sqrt{10}i$

This is the solution to our original equation, expressed in the standard complex number form $a ext{ extpm } b i$. Here, the real part $a$ is 3, and the imaginary part $b$ is $\sqrt{10}$. The $\pm$ symbol indicates two separate solutions: $x = 3 + \sqrt{10}i$ and $x = 3 - \sqrt{10}i$. Both are valid complex numbers.

Conclusion

We have successfully solved the equation $-(x-3)^2-3=7$ and expressed the answer in the standard complex number form $a ext{ extpm } b i$. The process involved isolating the squared term, taking the square root of both sides (which introduced the imaginary unit $i$ due to the negative value), and finally solving for $x$. The solutions are $x = 3 + \sqrt{10}i$ and $x = 3 - \sqrt{10}i$. This demonstrates how algebraic techniques can lead to solutions within the complex number system, which is crucial for a complete understanding of quadratic equations and beyond.

For further exploration into the fascinating world of complex numbers and their applications, you might find the resources at The Wolfram MathWorld website very helpful. They offer in-depth explanations and examples related to various mathematical concepts, including complex numbers.