Question

Solve the following quadratic equation using the perfect square trinomial pattern: 1/4x^2+2x+4=0
why is -4 the correct answer

Answer 1

To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we need to manipulate the equation into a form that fits \( (ax + b)^2 = c \).

First, let's multiply the entire equation by 4 to eliminate the fraction:

\[
4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 0
\]

This simplifies to:

\[
x^2 + 8x + 16 = 0
\]

Next, we can recognize that the left-hand side can be expressed as a perfect square:

\[
(x + 4)^2 = 0
\]

Now, to solve for \( x \), we can take the square root of both sides:

\[
x + 4 = 0
\]

Subtracting 4 from both sides gives:

\[
x = -4
\]

So, the solution to the equation is \( x = -4 \).

### Why is -4 the correct answer?

The answer \( x = -4 \) is correct because when we substitute it back into the original equation, it satisfies the equality:

1. Substitute \( x = -4 \) into the original equation:
\[
\frac{1}{4}(-4)^2 + 2(-4) + 4 = 0
\]
2. Calculate:
\[
\frac{1}{4} \cdot 16 - 8 + 4 = 0
\]
\[
4 - 8 + 4 = 0
\]
\[
0 = 0
\]

Since the left side equals the right side, it confirms that \( x = -4 \) is indeed the correct solution to the quadratic equation.