Find the roots of [tex]$f(x)=x^2+4 x+4$[/tex]
Answer (2)
Set the quadratic function equal to zero: x 2 + 4 x + 4 = 0 .
Factor the quadratic expression: ( x + 2 ) ( x + 2 ) = 0 .
Simplify the equation: ( x + 2 ) 2 = 0 .
Solve for x : x = − 2 . The root is − 2 .
Explanation
Understanding the Problem The problem asks us to find the roots of the function f ( x ) = x 2 + 4 x + 4 . In other words, we need to find the values of x for which f ( x ) = 0 .
Setting up the Equation To find the roots, we set the function equal to zero: x 2 + 4 x + 4 = 0
Factoring the Quadratic We can factor the quadratic expression as follows: ( x + 2 ) ( x + 2 ) = 0
Simplifying This simplifies to: ( x + 2 ) 2 = 0
Taking the Square Root Taking the square root of both sides, we get: x + 2 = 0
Solving for x Solving for x , we find: x = − 2
Final Answer Therefore, the function has a repeated root at x = − 2 .
Examples
Finding the roots of a quadratic equation is a fundamental concept in algebra with numerous real-world applications. For instance, engineers use quadratic equations to model the trajectory of a projectile, such as a ball thrown in the air or a rocket launched into space. The roots of the equation represent the points where the projectile hits the ground or reaches a certain altitude. Similarly, in business, quadratic equations can be used to model profit and cost functions, where the roots indicate break-even points. Understanding how to solve quadratic equations is essential for making informed decisions in various fields.
The root of the function f ( x ) = x 2 + 4 x + 4 is x = − 2 , which is a repeated root. The equation is set to zero, factored, and solved to find this solution. Therefore, the function touches the x-axis at x = − 2 without crossing it.
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