Question
Solve the following quadratic equation. Round to the nearest hundredth if necessary: (x+27)^2/-6 = -3
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(\frac{(x+27)^2}{-6} = -3\), we start by eliminating the fraction. Multiply both sides by \(-6\):
\[
(x+27)^2 = -3 \times -6
\]
This simplifies to:
\[
(x+27)^2 = 18
\]
Next, we take the square root of both sides:
\[
x + 27 = \pm \sqrt{18}
\]
Since \(\sqrt{18} = 3\sqrt{2}\), we can write:
\[
x + 27 = \pm 3\sqrt{2}
\]
Now, solving for \(x\):
1. For the positive case:
\[
x = -27 + 3\sqrt{2}
\]
2. For the negative case:
\[
x = -27 - 3\sqrt{2}
\]
Next, we calculate the approximate values for \(x\):
1. Calculate \(3\sqrt{2}\):
\[
\sqrt{2} \approx 1.414
\]
\[
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242
\]
2. For the positive case:
\[
x \approx -27 + 4.242 \approx -22.758
\]
Rounding to the nearest hundredth gives:
\[
x \approx -22.76
\]
3. For the negative case:
\[
x \approx -27 - 4.242 \approx -31.242
\]
Rounding to the nearest hundredth gives:
\[
x \approx -31.24
\]
Thus, the solutions to the equation are:
\[
\boxed{-22.76} \quad \text{and} \quad \boxed{-31.24}
\]
\[
(x+27)^2 = -3 \times -6
\]
This simplifies to:
\[
(x+27)^2 = 18
\]
Next, we take the square root of both sides:
\[
x + 27 = \pm \sqrt{18}
\]
Since \(\sqrt{18} = 3\sqrt{2}\), we can write:
\[
x + 27 = \pm 3\sqrt{2}
\]
Now, solving for \(x\):
1. For the positive case:
\[
x = -27 + 3\sqrt{2}
\]
2. For the negative case:
\[
x = -27 - 3\sqrt{2}
\]
Next, we calculate the approximate values for \(x\):
1. Calculate \(3\sqrt{2}\):
\[
\sqrt{2} \approx 1.414
\]
\[
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242
\]
2. For the positive case:
\[
x \approx -27 + 4.242 \approx -22.758
\]
Rounding to the nearest hundredth gives:
\[
x \approx -22.76
\]
3. For the negative case:
\[
x \approx -27 - 4.242 \approx -31.242
\]
Rounding to the nearest hundredth gives:
\[
x \approx -31.24
\]
Thus, the solutions to the equation are:
\[
\boxed{-22.76} \quad \text{and} \quad \boxed{-31.24}
\]
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