Solve $2 x^2-3 x+4=0$ by applying the quadratic formula.
Answer (1)
Identify the coefficients: a = 2 , b = − 3 , c = 4 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values and simplify: x = 4 3 ± − 23 = 4 3 ± 4 23 i .
The solutions are: x = 4 3 + 4 23 i , x = 4 3 − 4 23 i .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 − 3 x + 4 = 0 , and we need to solve it using the quadratic formula.
Quadratic Formula The quadratic formula 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 = − 3 , and c = 4 .
Substitution Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 2 ) − ( − 3 ) ± ( − 3 ) 2 − 4 ( 2 ) ( 4 ) x = 4 3 ± 9 − 32 x = 4 3 ± − 23
Simplification Since the discriminant (the value inside the square root) is negative, we will have complex solutions. We can rewrite − 23 as 23 i , where i is the imaginary unit ( i 2 = − 1 ). Therefore, x = 4 3 ± i 23 x = 4 3 ± 4 23 i
Final Solutions The two solutions are: x 1 = 4 3 + 4 23 i x 2 = 4 3 − 4 23 i Thus, the solutions to the quadratic equation 2 x 2 − 3 x + 4 = 0 are x = 4 3 + 4 23 i and x = 4 3 − 4 23 i .
Examples
Quadratic equations are used in various fields such as physics, engineering, and economics. For example, in physics, quadratic equations can be used to model the trajectory of a projectile. In engineering, they can be used to design parabolic reflectors. In economics, they can be used to model cost and revenue functions. Understanding how to solve quadratic equations is essential for solving real-world problems in these fields.
