To solve the equation \(x^2 + 3x = 24\) by completing the square, follow these steps:
1. **Move the constant term to the other side of the equation:**
\[ x^2 + 3x - 24 = 0. \]
2. **Isolate the terms involving \(x\):**
\[ x^2 + 3x = 24. \]
3. **Complete the square:**
- Take half of the coefficient of \(x\), which is 3, divide it by 2 to get \(\frac{3}{2}\), and then square it to get \(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\).
- Add and subtract \(\frac{9}{4}\) to/from the left-hand side of the equation:
\[ x^2 + 3x + \frac{9}{4} = 24 + \frac{9}{4}. \]
4. **Rewrite the left-hand side as a square of a binomial:**
\[ \left(x + \frac{3}{2}\right)^2 = 24 + \frac{9}{4}. \]
5. **Simplify the right-hand side:**
\[ \left(x + \frac{3}{2}\right)^2 = \frac{96}{4} + \frac{9}{4} = \frac{105}{4}. \]
6. **Take the square root of both sides:**
\[ x + \frac{3}{2} = \pm \sqrt{\frac{105}{4}}. \]
\[ x + \frac{3}{2} = \pm \frac{\sqrt{105}}{2}. \]
7. **Solve for \(x\):**
\[ x = -\frac{3}{2} \pm \frac{\sqrt{105}}{2}. \]
Which gives us:
\[ x = \frac{-3 \pm \sqrt{105}}{2}. \]
8. **Calculate the numerical values:**
- Using a calculator for the square root of 105:
\(\sqrt{105} \approx 10.246\).
So, the solutions are:
\[ x = \frac{-3 + 10.246}{2} \approx \frac{7.246}{2} = 3.623. \]
\[ x = \frac{-3 - 10.246}{2} \approx \frac{-13.246}{2} = -6.623. \]
Therefore, the rounded values to the nearest hundredth are approximately \(3.62\) and \(-6.62\).
So the correct answer is indeed:
**3.62, –6.62**
9.
Solve the equation by completing the square. Round to the nearest hundredth if necessary.
x2 + 3x = 24
3.62, –6.62
4.66, 5.12
3.55, –6.55
24.75, –27.75
1 answer