To solve the quadratic equation \( x^2 + x + 0.25 = 0 \) using the perfect square trinomial pattern, we can start by rewriting the left side of the equation.
The perfect square trinomial pattern can be expressed as: \[ (a + b)^2 = a^2 + 2ab + b^2 \]
In our case, we want to recognize \( x^2 + x + 0.25 \) as a perfect square. We can rewrite the equation as: \[ x^2 + x + \left(\frac{1}{2}\right)^2 = 0 \] Here, \( \left(\frac{1}{2}\right)^2 \) is \( 0.25 \), which gives us: \[ x^2 + x + \left(\frac{1}{2}\right)^2 = 0 \]
This can be factored into: \[ \left(x + \frac{1}{2}\right)^2 = 0 \]
Now, to solve for \( x \), we take the square root of both sides: \[ x + \frac{1}{2} = 0 \]
Subtracting \( \frac{1}{2} \) from both sides, we find: \[ x = -\frac{1}{2} \]
Thus, the only solution to the equation \( x^2 + x + 0.25 = 0 \) is: \[ \boxed{-0.5} \]
Regarding the options you provided:
- \( x = 0.25 \)
- \( x = -0.25 \)
- \( x = 0.5 \)
- \( x = -0.5 \)
The correct solution is \( x = -0.5 \).