Question
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Answer
The factored form of \( x^{2} + 2x + 1 \) is \( (x + 1)^2 \).
Solution
To factor the quadratic expression \( x^{2} + 2x + 1 \), we look for two numbers that:
1. **Add up to the coefficient of the middle term (2)**.
2. **Multiply to the constant term (1)**.
Let's identify these numbers:
- **Numbers that add up to 2**: 1 and 1.
- **Numbers that multiply to 1**: 1 and 1.
Both conditions are satisfied by the number 1. Therefore, the factored form of the quadratic expression is:
\[
x^{2} + 2x + 1 = (x + 1)(x + 1) = (x + 1)^2
\]
**Answer:**
The factored form is ( x + 1 ) squared, that is
\[
(x + 1)^2
\]
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Simplify this solution Bonus Knowledge
You can factor \( x^{2}+2x+1 \) into \( (x+1)(x+1) \) or simply write it as \( (x+1)^{2} \). This is a perfect square trinomial, where both roots are the same, which means the graph of this quadratic touches the x-axis at \( x = -1 \). It’s an excellent example of how some polynomials can be neatly packaged into a square! In practical terms, this factoring helps solve quadratic equations efficiently. For instance, if you set \( x^{2} + 2x + 1 = 0 \), you can quickly deduce that \( (x+1)^{2} = 0 \) leads to the solution \( x = -1 \). This not only speeds up your calculations but also provides deeper insights into the behavior of the function!
