Solve the quadratic equation numerically (using tables of [tex]$x$[/tex]- and [tex]$y$[/tex]-values). [tex]$x^2+2 x+1=0$[/tex]
A. [tex]$x=-1$[/tex]
B. [tex]$x=1$[/tex] or [tex]$x=-3$[/tex]
C. [tex]$x=-3$[/tex]
D. [tex]$x=2$[/tex] or [tex]$x=-1$[/tex]
Answer (2)
The quadratic equation x 2 + 2 x + 1 = 0 can be factored to ( x + 1 ) 2 = 0 , leading to the solution x = − 1 . Therefore, the correct answer is option A, x = − 1 .
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Factor the quadratic equation: x 2 + 2 x + 1 = ( x + 1 ) 2 .
Set the factor equal to zero: ( x + 1 ) = 0 .
Solve for x : x = − 1 .
The solution to the quadratic equation is x = − 1 .
Explanation
Analyze the problem We are given the quadratic equation x 2 + 2 x + 1 = 0 . Our goal is to solve this equation to find the value(s) of x that satisfy it. We can solve this equation by factoring, completing the square, or using the quadratic formula. In this case, factoring is the easiest method.
Factor the quadratic equation We can factor the quadratic expression as follows: x 2 + 2 x + 1 = ( x + 1 ) ( x + 1 ) = ( x + 1 ) 2 So, the equation becomes: ( x + 1 ) 2 = 0
Solve for x To find the roots, we set the factor equal to zero: x + 1 = 0 Solving for x , we get: x = − 1
Determine the solution Since both factors are the same, we have a repeated root. Therefore, the only solution to the quadratic equation is x = − 1 .
Select the correct answer Comparing our solution x = − 1 with the given options: a. x = − 1 b. x = 1 or x = − 3 c. x = − 3 d. x = 2 or x = − 1 Option a matches our solution.
Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area with a specific perimeter and area, and modeling the growth or decay of populations. For instance, if you throw a ball, its path can be modeled by a quadratic equation, allowing you to predict where it will land. Similarly, engineers use quadratic equations to design bridges and other structures, ensuring stability and safety.
