Sure, let's solve the given quadratic equation step-by-step to determine the correct solutions for \(x\).
1. Given Quadratic Equation:
[tex]\[
x^2 + 2x + 1 = 17
\][/tex]
2. Rearrange the equation to standard quadratic form \(ax^2 + bx + c = 0\):
[tex]\[
x^2 + 2x + 1 - 17 = 0
\][/tex]
Simplify the constant term:
[tex]\[
x^2 + 2x - 16 = 0
\][/tex]
3. Identify the coefficients \(a\), \(b\), and \(c\):
[tex]\[
a = 1, \quad b = 2, \quad c = -16
\][/tex]
4. Calculate the discriminant \(\Delta\) (which is \(b^2 - 4ac\)):
[tex]\[
\Delta = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-16)
\][/tex]
Calculate the values:
[tex]\[
\Delta = 4 + 64 = 68
\][/tex]
The discriminant \(\Delta\) is 68.
5. Using the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\):
Substitute \(a = 1\), \(b = 2\), and \(\Delta = 68\):
[tex]\[
x = \frac{-2 \pm \sqrt{68}}{2 \cdot 1}
\][/tex]
6. Simplify the solutions:
[tex]\[
x = \frac{-2 \pm \sqrt{68}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{4 \cdot 17}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm 2\sqrt{17}}{2}
\][/tex]
Simplify by dividing each term inside the fraction by 2:
[tex]\[
x = -1 \pm \sqrt{17}
\][/tex]
7. Final Solutions:
[tex]\[
x = -1 + \sqrt{17} \quad \text{and} \quad x = -1 - \sqrt{17}
\][/tex]
Therefore, the correct solution for the given equation \(x^2 + 2x + 1 = 17\) is:
[tex]\[
x = -1 \pm \sqrt{17}
\][/tex]
