Solve for $x$ in the equation $x^2+2 x+1=17$.
A. $x=-1 \pm \sqrt{15}$
B. $x=-1 \pm \sqrt{13}$
C. $x=-2 \pm 2 \sqrt{5}$
D. $x=-1 \pm \sqrt{17}$
Answer (1)
Recognize the left side as a perfect square: ( x + 1 ) 2 = 17 .
Take the square root of both sides: x + 1 = ± 17 .
Isolate x : x = − 1 ± 17 .
The solutions are x = − 1 ± 17 .
Explanation
Analyze the equation We are given the equation x 2 + 2 x + 1 = 17 and asked to solve for x . We can recognize that the left side of the equation is a perfect square.
Rewrite as a perfect square We can rewrite the equation as ( x + 1 ) 2 = 17 .
Take the square root Taking the square root of both sides, we get x + 1 = ± 17 .
Isolate x Subtracting 1 from both sides, we find x = − 1 ± 17 .
State the solutions Therefore, the solutions for x are x = − 1 + 17 and x = − 1 − 17 .
Examples
Completing the square is a useful technique in physics, such as when analyzing simple harmonic motion or projectile motion. For example, if you have an equation describing the position of an object as a function of time, completing the square can help you rewrite the equation in a form that reveals key information about the motion, such as the amplitude and phase shift. Suppose the height of a ball thrown vertically upwards is given by h ( t ) = − 5 t 2 + 20 t + 1 . By completing the square, we can find the maximum height and the time at which it occurs, which are important parameters in understanding the ball's trajectory.
