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Solve for [tex]$x[tex]$[/tex] in the equation [tex]$[/tex]x^2+2x+1=17$[/tex].

A. [tex]$x=-1 \pm \sqrt{15}$[/tex]
B. [tex]$x=-1 \pm \sqrt{17}$[/tex]
C. [tex]$x=-2 \pm 2\sqrt{5}$[/tex]
D. [tex]$x=-1 \pm \sqrt{13}$[/tex]


Sagot :

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]
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