Question

Solve the following quadratic equation using square roots:
−3+(x+12)^2/2=−3
(everything is the numerator over 2)
(1 point)
Type the smaller solution first.

Answer 1

To solve the equation

\[
-3 + \frac{(x + 12)^2}{2} = -3,
\]

we first simplify it by isolating the quadratic term. We can do this by adding 3 to both sides:

\[
\frac{(x + 12)^2}{2} = 0.
\]

Next, we multiply both sides by 2 to eliminate the fraction:

\[
(x + 12)^2 = 0.
\]

Now we take the square root of both sides:

\[
x + 12 = 0.
\]

To find \(x\), we subtract 12 from both sides:

\[
x = -12.
\]

Since the quadratic equation yields a double root, there is only one solution, which is

\[
x = -12.
\]

Thus, the answer is:

\[
\boxed{-12}.
\]

Answer 2

-3 + is apart of the numerator

Answer 3

Thank you for the clarification! Let's solve the equation again, ensuring that \(-3\) is in the numerator.

The equation you're looking at is:

\[
\frac{-3 + (x + 12)^2}{2} = -3.
\]

Let's start by getting rid of the denominator by multiplying both sides by 2:

\[
-3 + (x + 12)^2 = -6.
\]

Next, we'll isolate the quadratic term. We can add \(3\) to both sides:

\[
(x + 12)^2 = -6 + 3,
\]

which simplifies to:

\[
(x + 12)^2 = -3.
\]

Since the right side is negative, this means there are no real solutions, but we can find complex solutions by taking the square root of both sides:

\[
x + 12 = \pm \sqrt{-3}.
\]

We can rewrite this using imaginary numbers:

\[
x + 12 = \pm i\sqrt{3}.
\]

Now, we subtract 12 from both sides to solve for \(x\):

\[
x = -12 \pm i\sqrt{3}.
\]

Thus, the two complex solutions are:

\[
x = -12 + i \sqrt{3} \quad \text{and} \quad x = -12 - i \sqrt{3}.
\]

Since the problem asks for the smaller solution first, we can write it as:

\[
\boxed{-12 - i\sqrt{3}} \quad \text{(smaller solution)}
\]

and

\[
\boxed{-12 + i\sqrt{3}} \quad \text{(larger solution)}.
\]