Question
Solve the following quadratic equation using the perfect square trinormal pattern 1/4 x squared plus 2X + 4 = 0
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(\frac{1}{4}x^2 + 2x + 4 = 0\) using the perfect square trinomial pattern, we first want to rearrange or manipulate the equation into a perfect square.
1. **Multiply the entire equation by 4** to eliminate the fraction:
\[
4 \left(\frac{1}{4}x^2 + 2x + 4\right) = 4 \cdot 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
2. **Rewrite the left-hand side as a perfect square**:
Notice that \(x^2 + 8x + 16\) is a perfect square trinomial. It can be factored as:
\[
(x + 4)^2 = 0
\]
3. **Set the squared term equal to zero**:
\[
(x + 4)^2 = 0
\]
4. **Solve for \(x\)** by taking the square root of both sides:
\[
x + 4 = 0
\]
Subtract 4 from both sides:
\[
x = -4
\]
5. **Conclusion**:
The solution to the given quadratic equation \(\frac{1}{4}x^2 + 2x + 4 = 0\) is:
\[
\boxed{-4}
\]
1. **Multiply the entire equation by 4** to eliminate the fraction:
\[
4 \left(\frac{1}{4}x^2 + 2x + 4\right) = 4 \cdot 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
2. **Rewrite the left-hand side as a perfect square**:
Notice that \(x^2 + 8x + 16\) is a perfect square trinomial. It can be factored as:
\[
(x + 4)^2 = 0
\]
3. **Set the squared term equal to zero**:
\[
(x + 4)^2 = 0
\]
4. **Solve for \(x\)** by taking the square root of both sides:
\[
x + 4 = 0
\]
Subtract 4 from both sides:
\[
x = -4
\]
5. **Conclusion**:
The solution to the given quadratic equation \(\frac{1}{4}x^2 + 2x + 4 = 0\) is:
\[
\boxed{-4}
\]
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