Let's analyze Mariah's work and understand the nature of the solutions step-by-step.
Mariah started with the quadratic equation:
[tex]\[ x^2 + 4x + 1 = 0 \][/tex]
1. She rearranged it to:
[tex]\[ x^2 + 4x = -1 \][/tex]
2. Next, she completed the square:
[tex]\[ x^2 + 4x + 4 = -1 + 4 \][/tex]
3. This simplification led to:
[tex]\[ (x + 2)^2 = 3 \][/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x + 2 = \pm \sqrt{3} \][/tex]
Then solve for [tex]\( x \)[/tex]:
[tex]\[ x = -2 \pm \sqrt{3} \][/tex]
So, the solutions to the equation [tex]\( x^2 + 4x + 1 = 0 \)[/tex] are:
[tex]\[ x = -2 + \sqrt{3} \quad \text{and} \quad x = -2 - \sqrt{3} \][/tex]
Now, let's analyze the nature of these solutions:
- [tex]\(-2\)[/tex] is a rational number.
- [tex]\(\sqrt{3}\)[/tex] is an irrational number.
For the sum and difference of these types of numbers:
- The sum of a rational number and an irrational number is always irrational.
- The difference between a rational number and an irrational number is always irrational.
Thus, the solutions [tex]\(-2 + \sqrt{3}\)[/tex] and [tex]\(-2 - \sqrt{3}\)[/tex] are both irrational because they consist of a rational part ([tex]\(-2\)[/tex]) and an irrational part ([tex]\(\sqrt{3}\)[/tex]).
Hence, the correct description of the solutions is:
Since -2 is rational and [tex]\(\sqrt{3}\)[/tex] is irrational, the sum and difference are irrational.
