Find all real solutions. (Enter your answers as comma-separated lists. If there is no real solution, enter NO REALSOLUTION.)
[tex]
\begin{array}{l}
2 x^2+4 x+1=0 \\
x=?
\end{array}
[/tex]
Answer (1)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 2 , b = 4 , and c = 1 .
Substitute the values into the formula: x = 2 ( 2 ) − 4 ± 4 2 − 4 ( 2 ) ( 1 ) .
Simplify the expression: x = 4 − 4 ± 2 2 .
Obtain the two real solutions: x = − 1 + 2 2 , x = − 1 − 2 2 .
The solutions are: − 1 + 2 2 , − 1 − 2 2
Explanation
Understanding the Problem We are given a quadratic equation 2 x 2 + 4 x + 1 = 0 . Our goal is to find all real solutions for x . We can use the quadratic formula to solve this equation.
Applying the Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c for a quadratic equation of the form a x 2 + b x + c = 0 . In this case, a = 2 , b = 4 , and c = 1 .
Substitution Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 2 ) − 4 ± 4 2 − 4 ( 2 ) ( 1 )
Simplification Simplify the expression: x = 4 − 4 ± 16 − 8 = 4 − 4 ± 8 x = 4 − 4 ± 2 2 x = − 1 ± 2 2
Finding the Solutions The two real solutions are x = − 1 + 2 2 and x = − 1 − 2 2 . Approximating these values, we get x ≈ − 0.29289 and x ≈ − 1.70711 .
Final Answer Therefore, the real solutions to the quadratic equation 2 x 2 + 4 x + 1 = 0 are x = − 1 + 2 2 and x = − 1 − 2 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 given its 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 a perimeter of 40 meters, you can use a quadratic equation to find the length and width of the garden. Understanding how to solve quadratic equations is essential for solving many practical problems in engineering, physics, and economics.
