Use the quadratic formula to solve the equation.
[tex]2 x^2-3 x+8=0[/tex]
Answer (1)
Identify coefficients: a = 2 , b = − 3 , c = 8 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 4 3 ± 9 − 64 = 4 3 ± − 55 .
Express with imaginary unit: x = 4 3 ± i 55 .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 − 3 x + 8 = 0 . Our goal is to solve for x 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 .
Identifying Coefficients In our equation, 2 x 2 − 3 x + 8 = 0 , we identify the coefficients as a = 2 , b = − 3 , and c = 8 .
Substituting Values Now, we substitute these values into the quadratic formula: x = 2 ( 2 ) − ( − 3 ) ± ( − 3 ) 2 − 4 ( 2 ) ( 8 )
Simplifying Simplify the expression: x = 4 3 ± 9 − 64 x = 4 3 ± − 55
Introducing Imaginary Unit Since we have a negative number under the square root, we introduce the imaginary unit i , where i = − 1 . Thus, − 55 = i 55 .
x = 4 3 ± i 55
Final Answer Therefore, the solutions to the quadratic equation are: x = 4 3 + i 55 and x = 4 3 − i 55 So the correct answer is 4 3 ± i 55 .
Examples
Quadratic equations are incredibly useful in physics, engineering, and economics. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the arches, ensuring structural integrity and optimal load distribution. Similarly, in finance, quadratic equations can help model investment growth or calculate break-even points for businesses. Understanding how to solve these equations is a fundamental skill that enables professionals to make informed decisions and create efficient designs.
