Solve the equation using the quadratic formula.
[tex]4 x^2=2 x-3[/tex]
The solution set is { }. (Simplify your answer, including any radicals)
Answer (2)
Rewrite the given equation 4 x 2 = 2 x − 3 in the standard form: 4 x 2 − 2 x + 3 = 0 .
Identify the coefficients: a = 4 , b = − 2 , and c = 3 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 4 ) 2 ± ( − 2 ) 2 − 4 ( 4 ) ( 3 ) .
Simplify the expression to find the complex solutions: x = 4 1 ± i 11 , so the solution set is { 4 1 − i 11 , 4 1 + i 11 } .
Explanation
Understanding the Problem We are given the quadratic equation 4 x 2 = 2 x − 3 . Our goal is to solve this equation using the quadratic formula. The quadratic formula is a general method for finding the solutions (also called roots) of any quadratic equation in the form a x 2 + b x + c = 0 .
Rewriting the Equation First, we need to rewrite the given equation in the standard form a x 2 + b x + c = 0 . Subtracting 2 x and adding 3 to both sides of the equation 4 x 2 = 2 x − 3 , we get
4 x 2 − 2 x + 3 = 0
Identifying Coefficients Now, we can identify the coefficients a , b , and c in the standard form a x 2 + b x + c = 0 . In our equation 4 x 2 − 2 x + 3 = 0 , we have:
a = 4 b = − 2 c = 3
Stating the Quadratic Formula The quadratic formula is given by:
x = 2 a − b ± b 2 − 4 a c
Substituting Values Now, we substitute the values of a , b , and c into the quadratic formula:
x = 2 ( 4 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 4 ) ( 3 )
x = 8 2 ± 4 − 48
x = 8 2 ± − 44
Simplifying the Expression Since the discriminant (the value inside the square root) is negative, we will have complex solutions. We can simplify the square root of − 44 as follows:
− 44 = 4 × − 11 = 4 × − 11 = 2 − 11 = 2 i 11
So, we have:
x = 8 2 ± 2 i 11
x = 4 1 ± i 11
Final Answer Therefore, the two solutions are:
x = 4 1 + i 11 and x = 4 1 − i 11
Thus, the solution set is { 4 1 + i 11 , 4 1 − i 11 } .
Examples
The quadratic formula is not just an abstract mathematical tool; it has practical applications in various fields. For example, in physics, when analyzing projectile motion under constant acceleration, the quadratic formula can be used to determine the time it takes for an object to reach a certain height. Similarly, in engineering, it can be used to calculate the dimensions of structures or electrical circuits to meet specific performance criteria. Understanding and applying the quadratic formula allows us to solve real-world problems involving quadratic relationships.
To solve the equation 4 x 2 = 2 x − 3 , we rewrite it in standard form as 4 x 2 − 2 x + 3 = 0 and apply the quadratic formula. This yields complex solutions, which simplify to x = 4 1 ± i 11 . The solution set is { 4 1 + i 11 , 4 1 − i 11 } .
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