Solve for $x$ in the equation $x^2+2 x+1=17$

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

Recognize the left side as a perfect square: ( x + 1 ) 2 = 17 .
Take the square root of both sides: x + 1 = ± 17 ​ .
Isolate x by subtracting 1: x = − 1 ± 17 ​ .
The solutions are x = − 1 + 17 ​ and x = − 1 − 17 ​ , thus x = − 1 ± 17 ​ ​ .

Explanation

Problem Analysis We are given the equation x 2 + 2 x + 1 = 17 . Our goal is to solve for x .

Rewriting the Equation Notice that the left side of the equation is a perfect square: x 2 + 2 x + 1 = ( x + 1 ) 2 . So we can rewrite the equation as ( x + 1 ) 2 = 17 .

Taking the Square Root Now, we take the square root of both sides of the equation: ( x + 1 ) 2 ​ = ± 17 ​ This simplifies to x + 1 = ± 17 ​ .

Solving for x Finally, we solve for x by subtracting 1 from both sides: x = − 1 ± 17 ​ So the solutions are x = − 1 + 17 ​ and x = − 1 − 17 ​ .


Examples
Imagine you are designing a square garden and want to increase its area to 17 square meters by adding the same length to two adjacent sides. If the original side length plus the added length is represented by x , and the increase in area is modeled by the equation ( x + 1 ) 2 = 17 , solving this equation will give you the required length to add to each side to achieve the desired area. This problem demonstrates how quadratic equations can be used in practical design and measurement scenarios.

Answered by GinnyAnswer | 2025-07-03

We solved the equation x 2 + 2 x + 1 = 17 by recognizing it as a perfect square and taking the square root, leading us to the solution x = − 1 ± 17 ​ . Therefore, the correct answer is option B.
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Answered by Anonymous | 2025-07-04