Solve $2 x^2+4 x+10=0$
Answer (2)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 2 , b = 4 , and c = 10 .
Calculate the discriminant: b 2 − 4 a c = 4 2 − 4 ( 2 ) ( 10 ) = − 64 .
Simplify the square root of the discriminant: − 64 = 8 i .
Find the solutions: x = 4 − 4 ± 8 i = − 1 ± 2 i . The final answer is − 1 ± 2 i .
Explanation
Understanding the Problem We are asked to solve the quadratic equation 2 x 2 + 4 x + 10 = 0 . This means we need to find the values of x that satisfy this equation. We will use the quadratic formula to find these values.
Identifying the 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 = 4 , and c = 10 .
Substituting the Values Now, we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 2 ) − 4 ± 4 2 − 4 ( 2 ) ( 10 )
Simplifying the Discriminant Next, we simplify the expression under the square root: 4 2 − 4 ( 2 ) ( 10 ) = 16 − 80 = − 64
Dealing with Complex Numbers Since the discriminant (the value under the square root) is negative, we will have complex solutions. We rewrite the expression as: x = 4 − 4 ± − 64 We know that − 64 = 64 ⋅ − 1 = 8 i , where i is the imaginary unit ( i 2 = − 1 ).
Simplifying the Solution Substitute this back into the equation: x = 4 − 4 ± 8 i Now, we divide both the real and imaginary parts by 4 to simplify the expression: x = − 1 ± 2 i
Final Answer Therefore, the solutions to the quadratic equation 2 x 2 + 4 x + 10 = 0 are x = − 1 + 2 i and x = − 1 − 2 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. When analyzing alternating current (AC) circuits, complex numbers help represent the impedance, which is the opposition to the flow of current. By using complex impedance, engineers can easily calculate the voltage and current in AC circuits, design filters, and ensure stable power delivery. This makes the design and analysis of electrical systems much more efficient.
The solutions to the quadratic equation 2 x 2 + 4 x + 10 = 0 are complex numbers: x = − 1 + 2 i and x = − 1 − 2 i . We calculated this using the quadratic formula, which involves finding the discriminant, and since it was negative, the solutions contain imaginary parts. This shows how complex numbers are useful in solving equations that do not have real solutions.
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