Solve the equation −(x−3)^2−3 =7 and re-express the answer as a complex number in the standard from a±bi .(1 point)

Question

GPT-4o mini · Answer

To solve the equation $- (x - 3)^2 - 3 = 7$, we can start by isolating the quadratic term. 1. Add 3 to both sides of the equation: $$ - (x - 3)^2 = 7 + 3 $$ $$ - (x - 3)^2 = 10 $$ 2. Multiply both sides by -1: $$ (x - 3)^2 = -10 $$ 3. Now, take the square root of both sides: $$ x - 3 = \pm \sqrt{-10} $$ Using the fact that $\sqrt{-10} = i\sqrt{10}$ (where $i = \sqrt{-1}$): $$ x - 3 = \pm i\sqrt{10} $$ 4. Solve for $x$: $$ x = 3 \pm i\sqrt{10} $$ Thus, the solutions are: $$ x = 3 + i\sqrt{10} \quad \text{and} \quad x = 3 - i\sqrt{10} $$ Expressing the answer in the standard form $a + bi$: 1. For $x = 3 + i\sqrt{10}$, we have $a = 3$ and $b = \sqrt{10}$. 2. For $x = 3 - i\sqrt{10}$, we have $a = 3$ and $b = -\sqrt{10}$. So the solutions in the standard form $a \pm bi$ are: $$ x = 3 + i\sqrt{10} \quad \text{and} \quad x = 3 - i\sqrt{10} $$

sayletyler · Answer

Its a blank what would you like me to put not mutiple choice

GPT-4o mini · Answer

If you're looking for a single answer to fill in the blank, you can write it in the standard form $a + bi$ or $a - bi$. Here's what you can put in the blank:

For the first solution: \[ 3 + i\sqrt{10} \]
For the second solution: \[ 3 - i\sqrt{10} \]

You can choose either of these two solutions to fill in the blank. If only one entry is allowed, you could go with:

\[ 3 + i\sqrt{10} \]

Or, if your context specifies to prefer the negative imaginary part, you could use:

\[ 3 - i\sqrt{10} \]

Just pick one of these forms depending on what is required.