To solve the equation by completing the square, let's start with the given quadratic equation:
\[ x^2 - 3x - 12 = 0 \]
### Step-by-Step Solution:
1. **Move the constant term to the other side:**
\[ x^2 - 3x = 12 \]
2. **Complete the square:**
To complete the square, we need to form a perfect square trinomial on the left-hand side. To do this, we take the coefficient of \( x \) (which is -3), divide it by 2, and square the result:
\[
\left( \frac{-3}{2} \right)^2 = \left( -\frac{3}{2} \right)^2 = \frac{9}{4}
\]
3. **Add and subtract this square inside the equation:**
\[
x^2 - 3x + \frac{9}{4} = 12 + \frac{9}{4}
\]
4. **Simplify the equation:**
\[
x^2 - 3x + \frac{9}{4} = 12 + \frac{9}{4}
\]
Convert 12 to a fraction with a common denominator:
\[
12 = \frac{48}{4}
\]
So,
\[
12 + \frac{9}{4} = \frac{48}{4} + \frac{9}{4} = \frac{57}{4}
\]
5. **Rewrite the left-hand side as a perfect square:**
\[
\left( x - \frac{3}{2} \right)^2 = \frac{57}{4}
\]
6. **Solve for \( x \) by taking the square root of both sides:**
\[
x - \frac{3}{2} = \pm \sqrt{\frac{57}{4}}
\]
Simplify the right-hand side:
\[
x - \frac{3}{2} = \pm \frac{\sqrt{57}}{2}
\]
7. **Solve for \( x \):**
\[
x = \frac{3}{2} \pm \frac{\sqrt{57}}{2}
\]
8. **Combine the fractions:**
\[
x = \frac{3 \pm \sqrt{57}}{2}
\]
### Finding the approximate values:
Using a calculator to approximate \(\sqrt{57}\):
\[
\sqrt{57} \approx 7.55
\]
Substitute this approximation back into the equation:
\[
x = \frac{3 \pm 7.55}{2}
\]
This gives us two solutions:
1. \( x = \frac{3 + 7.55}{2} = \frac{10.55}{2} = 5.275 \)
2. \( x = \frac{3 - 7.55}{2} = \frac{-4.55}{2} = -2.275 \)
Rounding to the nearest hundredth, the solutions are:
\[
x \approx 5.27 \quad \text{and} \quad x \approx -2.27
\]
Thus, the correct answer is:
\[ \boxed{5.27, -2.27} \]
14.
Solve the equation by completing the square. Round to the nearest hundredth if necessary.
x2 – 3x – 12 = 0
3.12, 3.77
5.27, –2.27
–12.75, 15.75
4.74, –1.74
1 answer