Solve by factoring or finding square roots.
[tex]x^2+2 x+1=0[/tex]
Answer (1)
Factor the quadratic expression: x 2 + 2 x + 1 = ( x + 1 ) 2 .
Rewrite the equation: ( x + 1 ) 2 = 0 .
Take the square root of both sides: x + 1 = 0 .
Solve for x : x = − 1 . The solution is − 1 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 2 x + 1 = 0 . Our goal is to solve for x by either factoring or finding square roots.
Factoring the Quadratic Notice that the left-hand side of the equation is a perfect square trinomial. We can factor it as follows: x 2 + 2 x + 1 = ( x + 1 ) ( x + 1 ) = ( x + 1 ) 2
Rewriting the Equation Now we can rewrite the equation as: ( x + 1 ) 2 = 0
Taking the Square Root To solve for x , we take the square root of both sides of the equation: ( x + 1 ) 2 = 0 ∣ x + 1∣ = 0 x + 1 = 0
Solving for x Finally, we solve for x :
x = − 1
Final Answer Therefore, the solution to the quadratic equation x 2 + 2 x + 1 = 0 is x = − 1 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and you want the length to be 5 meters longer than the width, you can set up a quadratic equation to find the dimensions of the garden. Factoring and solving quadratic equations helps in optimizing designs and predicting outcomes in many practical situations.
