Solve the following equation: [tex]$4 x^2+40 x+100=0$[/tex]
Answer (2)
Divide the equation by 4 to simplify: x 2 + 10 x + 25 = 0 .
Recognize the perfect square trinomial: ( x + 5 ) 2 = 0 .
Take the square root of both sides: x + 5 = 0 .
Solve for x : x = − 5 .
Explanation
Problem Analysis We are given the quadratic equation 4 x 2 + 40 x + 100 = 0 . Our goal is to find the value(s) of x that satisfy this equation.
Simplifying the Equation First, we can simplify the equation by dividing all terms by 4: 4 4 x 2 + 4 40 x + 4 100 = 4 0 x 2 + 10 x + 25 = 0
Factoring the Quadratic Now, we recognize that the left side of the equation is a perfect square trinomial. It can be factored as: ( x + 5 ) ( x + 5 ) = 0 Or, equivalently: ( x + 5 ) 2 = 0
Solving for x To solve for x , we take the square root of both sides of the equation: ( x + 5 ) 2 = 0 x + 5 = 0
The Final Answer Finally, we isolate x by subtracting 5 from both sides: x = − 5
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, and modeling growth and decay processes. For instance, if you want to build a rectangular garden with an area of 100 square meters and you know that the length must be 5 meters more than the width, you can set up a quadratic equation to find the dimensions of the garden. Understanding how to solve quadratic equations allows you to solve these types of practical problems.
The equation 4 x 2 + 40 x + 100 = 0 simplifies to x 2 + 10 x + 25 = 0 by dividing by 4. This factors to ( x + 5 ) 2 = 0 , leading to the solution x = − 5 .
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