(b) [tex]2 x^2+4 x+1=0[/tex]
[tex]x=[/tex]
Answer (1)
Identify the coefficients: a = 2 , b = 4 , and c = 1 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values and simplify: x = 4 − 4 ± 2 2 .
Obtain the solutions: x = 2 − 2 ± 2 , so x = 2 − 2 + 2 and x = 2 − 2 − 2 .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 + 4 x + 1 = 0 and asked to find the values of x that satisfy this equation.
Applying the Quadratic Formula We can solve this quadratic equation using the quadratic formula, which is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 2 , b = 4 , and c = 1 .
Substitution Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 2 ) − 4 ± 4 2 − 4 ( 2 ) ( 1 )
Simplification Now, we simplify the expression: x = 4 − 4 ± 16 − 8 x = 4 − 4 ± 8 Since 8 = 4 × 2 = 2 2 , we have: x = 4 − 4 ± 2 2
Final Solutions We can divide both the numerator and the denominator by 2 to further simplify: x = 2 − 2 ± 2 Thus, the two solutions for x are: x = 2 − 2 + 2 ≈ − 0.2929 and x = 2 − 2 − 2 ≈ − 1.7071
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 given perimeter and area, or modeling the growth of a population. For instance, 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.
