Question

x^2+x-5=0 by completing the square

Answer 1

To solve the equation \( x^2 + x - 5 = 0 \) by completing the square, follow these steps:

1. Start with the original equation:
\[
x^2 + x - 5 = 0
\]

2. Move the constant to the other side:
\[
x^2 + x = 5
\]

3. To complete the square, you need to take the coefficient of \( x \), divide it by 2, and square the result. The coefficient of \( x \) is 1. So, we calculate:
\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]

4. Add \( \frac{1}{4} \) to both sides of the equation:
\[
x^2 + x + \frac{1}{4} = 5 + \frac{1}{4}
\]

5. Simplify the right side:
\[
5 = \frac{20}{4} \quad \Rightarrow \quad 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}
\]

6. The left side can now be factored:
\[
\left( x + \frac{1}{2} \right)^2 = \frac{21}{4}
\]

7. Take the square root of both sides:
\[
x + \frac{1}{2} = \pm \sqrt{\frac{21}{4}}
\]
Simplifying the square root gives:
\[
x + \frac{1}{2} = \pm \frac{\sqrt{21}}{2}
\]

8. Solve for \( x \) by isolating it:
\[
x = -\frac{1}{2} \pm \frac{\sqrt{21}}{2}
\]

9. Simplifying gives the two possible solutions:
\[
x = \frac{-1 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{21}}{2}
\]

Thus, the solutions to the equation \( x^2 + x - 5 = 0 \) are:
\[
x = \frac{-1 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{21}}{2}
\]