Find all real and complex solutions of the equation [tex]$2 x^2+4 x+3=0$[/tex]. (Enter your answers as a comma-separated list.)
Answer (1)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 2 , b = 4 , and c = 3 .
Simplify the expression to x = 4 − 4 ± − 8 .
Express the square root of -8 as 2 i 2 , leading to x = 4 − 4 ± 2 i 2 .
Divide by 4 to obtain the solutions: x = − 1 ± 2 i 2 , thus the answer is − 1 + 2 i 2 , − 1 − 2 i 2 .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 + 4 x + 3 = 0 . Our goal is to find all real and complex solutions for x . We will use the quadratic formula to solve this equation.
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 our case, we have a = 2 , b = 4 , and c = 3 .
Substitution Substituting the values of a , b , and c into the quadratic formula, we get:
x = 2 ( 2 ) − 4 ± 4 2 − 4 ( 2 ) ( 3 )
Simplification Now, we simplify the expression:
x = 4 − 4 ± 16 − 24
x = 4 − 4 ± − 8
Complex Numbers Since we have a negative number under the square root, we will have complex solutions. We can express − 8 as 8 i = 2 2 i . Thus,
x = 4 − 4 ± 2 i 2
Final Solutions Dividing both terms in the numerator by 4, we get:
x = − 1 ± 2 i 2
So the two solutions are x = − 1 + 2 i 2 and x = − 1 − 2 i 2 .
Conclusion Therefore, the solutions to the equation 2 x 2 + 4 x + 3 = 0 are x = − 1 + 2 i 2 and x = − 1 − 2 i 2 .
Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For instance, when designing a parabolic mirror for a telescope, engineers use quadratic equations to determine the precise curvature needed to focus light correctly. Similarly, in physics, projectile motion is described by quadratic equations, allowing us to predict the trajectory of a ball thrown in the air or the path of a rocket. Understanding how to solve these equations is crucial for accurate calculations and designs in these fields.
