Solve the equation using the quadratic formula. [tex]4 x^2-x=-2[/tex]. The solution set is (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
Answer (2)
Rewrite the equation in standard form: 4 x 2 − x + 2 = 0 .
Identify the coefficients: a = 4 , b = − 1 , c = 2 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 8 1 ± − 31 .
Express the solution set with complex numbers: { 8 1 + 8 31 i , 8 1 − 8 31 i } .
Explanation
Rewrite the equation First, we need to rewrite the given equation in the standard quadratic form, which is a x 2 + b x + c = 0 . The given equation is 4 x 2 − x = − 2 . Adding 2 to both sides, we get 4 x 2 − x + 2 = 0 .
Identify coefficients Now, we identify the coefficients a , b , and c . In the equation 4 x 2 − x + 2 = 0 , we have a = 4 , b = − 1 , and c = 2 .
Apply quadratic formula Next, we apply the quadratic formula, which is given by x = 2 a − b ± b 2 − 4 a c . Substituting the values of a , b , and c , we get:
x = 2 ( 4 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 4 ) ( 2 )
x = 8 1 ± 1 − 32
x = 8 1 ± − 31
Since the discriminant is negative, we have complex solutions.
Simplify the expression We can rewrite the square root of -31 as − 31 = 31 i , where i is the imaginary unit ( i 2 = − 1 ). Therefore,
x = 8 1 ± i 31
x = 8 1 ± 8 31 i
State the solution set Thus, the two solutions are x = 8 1 + 8 31 i and x = 8 1 − 8 31 i . The solution set is { 8 1 + 8 31 i , 8 1 − 8 31 i } .
Examples
The quadratic formula is not just an abstract concept; it has real-world applications. For instance, engineers use it to calculate the stability of bridges and buildings, ensuring they can withstand various forces. Similarly, in physics, it helps determine the trajectory of projectiles, like rockets or even a ball thrown in the air. By understanding the quadratic formula, you're unlocking tools that are essential in many fields that shape our world.
The solutions to the equation 4 x 2 − x = − 2 are complex and can be expressed as x = 8 1 + 8 31 i and x = 8 1 − 8 31 i . Thus, the solution set is { 8 1 + 8 31 i , 8 1 − 8 31 i } .
;
