solve the following quadratic equation. Round to the nearest hundredth if necessary: (x+27)^2/-6 = -3

Question

enter the smaller of the 2 values first.

Answer 1

To solve the equation \((x + 27)^2 / -6 = -3\), we first eliminate the fraction by multiplying both sides by \(-6\):

\[ (x + 27)^2 = -3 \times -6 \]

\[ (x + 27)^2 = 18 \]

Next, we take the square root of both sides:

\[ x + 27 = \pm \sqrt{18} \]

We can simplify \(\sqrt{18}\):

\[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

Now we express the two cases for \(x + 27\):

1. \(x + 27 = 3\sqrt{2}\)
2. \(x + 27 = -3\sqrt{2}\)

Next, we will solve for \(x\) in each case:

1. \(x = 3\sqrt{2} - 27\)
2. \(x = -3\sqrt{2} - 27\)

Now, we can compute \(3\sqrt{2}\):

\[ 3\sqrt{2} \approx 3 \times 1.414 \approx 4.242 \]

Thus,

1. \(x \approx 4.242 - 27 \approx -22.758\)
2. \(x \approx -4.242 - 27 \approx -31.242\)

Now, rounding to the nearest hundredth:

1. \(x_1 \approx -22.76\)
2. \(x_2 \approx -31.24\)

The smaller of the two values is:

\[ \boxed{-31.24} \]

And the other value is:

\[ \boxed{-22.76} \]

So, the final answer with the smaller value first is:

\[ \boxed{-31.24, -22.76} \]