Ct substitution of $a, b$, and $c$ in the
$x=\frac{12 \cdot \sqrt{12-114(1)}}{2(4)}$
Answer (2)
Simplify the expression inside the square root: 12 − 114 ( 1 ) = − 102 .
Substitute the simplified value back into the expression: x = 2 ( 4 ) 12 ⋅ − 102 .
Simplify the fraction: x = 2 3 ⋅ − 102 .
Express the square root of a negative number using the imaginary unit i : x = 2 3 102 i .
The final simplified expression is: 2 3 102 i .
Explanation
Understanding the Problem We are given the expression x = 2 ( 4 ) 12 ⋅ 12 − 114 ( 1 ) and we need to simplify it to find the value of x .
Simplifying the Expression Inside the Square Root First, simplify the expression inside the square root: 12 − 114 ( 1 ) = 12 − 114 = − 102
Substituting the Value Back into the Expression Substitute this value back into the expression: x = 2 ( 4 ) 12 ⋅ − 102 = 8 12 ⋅ − 102
Simplifying the Fraction Simplify the fraction: x = 2 3 ⋅ − 102
Expressing the Square Root of a Negative Number Express the square root of a negative number using the imaginary unit i :
− 102 = 102 i
Final Answer Substitute this into the expression: x = 2 3 102 i So, the final answer is: x = 2 3 102 i We can approximate 102 as approximately 10.1, so x ≈ 2 3 ⋅ 10.1 i ≈ 15.15 i
Examples
Complex numbers are useful in electrical engineering, particularly in analyzing AC circuits. The impedance of a circuit, which is the opposition to the flow of current, can be represented as a complex number. By using complex numbers, engineers can easily calculate the voltage and current in AC circuits, which is crucial for designing and troubleshooting electrical systems. This allows for efficient and reliable operation of various electronic devices and power systems.
The expression x = 2 ( 4 ) 12 ⋅ 12 − 114 ( 1 ) simplifies to x = 2 3 102 i , which shows that it involves the imaginary unit due to the negative number under the square root. The approximate value of x can be calculated as 15.15 i .
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