Sure, let's solve the equation [tex]\(x^2 + 2x + 1 = 17\)[/tex] step by step.
### Step 1: Rewrite the equation
First, we need to set the equation to zero by moving 17 to the left side:
[tex]\[ x^2 + 2x + 1 - 17 = 0 \][/tex]
### Step 2: Simplify the equation
This simplifies to:
[tex]\[ x^2 + 2x - 16 = 0 \][/tex]
### Step 3: Identify coefficients
The equation is now in the form of a standard quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -16\)[/tex]
### Step 4: Use the quadratic formula
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 5: Calculate the discriminant
The discriminant ([tex]\(\Delta\)[/tex]) is calculated as:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ \Delta = 2^2 - 4 \cdot 1 \cdot (-16) \][/tex]
[tex]\[ \Delta = 4 + 64 \][/tex]
[tex]\[ \Delta = 68 \][/tex]
### Step 6: Calculate the roots
Now that we have the discriminant, let's find the solutions for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{68}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{68}}{2} \][/tex]
Simplify inside the square root:
[tex]\[ \sqrt{68} = \sqrt{4 \cdot 17} = 2\sqrt{17} \][/tex]
Substitute back:
[tex]\[ x = \frac{-2 \pm 2\sqrt{17}}{2} \][/tex]
Simplify the expression:
[tex]\[ x = -1 \pm \sqrt{17} \][/tex]
### Step 7: Conclusion
Therefore, the solutions to the equation [tex]\(x^2 + 2x + 1 = 17\)[/tex] are:
[tex]\[ x = -1 + \sqrt{17} \][/tex]
[tex]\[ x = -1 - \sqrt{17} \][/tex]
Given the choices in the question, none of them match exactly, but based on our calculations, the correct form of the solutions are [tex]\( \boxed{-1 \pm \sqrt{17}} \)[/tex].
