Solve the following equation: [tex]$9 x^2+36 x+36=0$[/tex]
Answer (2)
To solve the equation 9 x 2 + 36 x + 36 = 0 , divide it by 9 to simplify to x 2 + 4 x + 4 = 0 . Factor the equation to get ( x + 2 ) 2 = 0 , leading to the solution x = − 2 . The equation has a double root at this value, meaning the graph touches the x-axis at this point.
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Divide the equation by 9 to simplify: x 2 + 4 x + 4 = 0 .
Factor the quadratic equation: ( x + 2 ) 2 = 0 .
Take the square root of both sides: x + 2 = 0 .
Solve for x: x = − 2 . The solution to the equation is − 2 .
Explanation
Problem Analysis We are given the quadratic equation 9 x 2 + 36 x + 36 = 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 9: 9 9 x 2 + 9 36 x + 9 36 = 9 0 x 2 + 4 x + 4 = 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 + 2 ) ( x + 2 ) = 0 Or, equivalently: ( x + 2 ) 2 = 0
Taking the Square Root To solve for x , we take the square root of both sides of the equation: ( x + 2 ) 2 = 0 x + 2 = 0
Solving for x Finally, we isolate x by subtracting 2 from both sides: x = − 2
Final Answer Therefore, the solution to the quadratic equation 9 x 2 + 36 x + 36 = 0 is x = − 2 .
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 with a specific perimeter and area, 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 know that one side must be 5 meters longer than the other, you can set up a quadratic equation to find the dimensions of the garden. Understanding how to solve quadratic equations helps in optimizing designs and predicting outcomes in many practical situations.
