Question

solve the following quadratic equation using the perfect square trinomial pattern x^2+x+0.25=0
x=0.25
[x=-0.25
x=0.5
x=-0.5

Answer 1

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 \( x = -0.5 \).