\( 3 x ^ { 2 } + 5 x + 3 = 0 \) .What is the solution(s)? Give your exact solution(s) in standard form of a complex number, no rounding. How can you make sure you have the correct number and type of solutions?
Pregunta
Answer
\(3 x ^ { 2 } + 5 x + 3 = 0 : x = - \frac { 5 } { 6 } + i \frac { \sqrt { 11 } } { 6 } , x = - \frac { 5 } { 6 } - i \frac { \sqrt { 11 } } { 6 } \)
Steps
\( 3 x ^ { 2 } + 5 x + 3 = 0\)
Solve with the quadratic formula
\( x _ { 1,2 } = \frac { - 5 \pm \sqrt { 5 ^ { 2 } - 4 \cdot 3 \cdot 3 } } { 2 \cdot 3 } \)
Simplify \( \sqrt { 5 ^ { 2 } - 4 \cdot 3 \cdot 3 } : \sqrt { 11 } i\)
\(x _ { 1,2 } = \frac { - 5 \pm \sqrt { 11 } i } { 2 \cdot 3 } \)
Separate the solutions
\( x _ { 1 } = \frac { - 5 + \sqrt { 11 } i } { 2 \cdot 3 } , x _ { 2 } = \frac { - 5 - \sqrt { 11 } i } { 2 \cdot 3 }\)
\(x = \frac { - 5 + \sqrt { 11 } i } { 2 \cdot 3 } : - \frac { 5 } { 6 } + i \frac { \sqrt { 11 } } { 6 } \)
\( x = \frac { - 5 - \sqrt { 11 } i } { 2 \cdot 3 } : - \frac { 5 } { 6 } - i \frac { \sqrt { 11 } } { 6 } \)
The solutions to the quadratic equation are:
\( x = - \frac { 5 } { 6 } + i \frac { \sqrt { 11 } } { 6 } , x = - \frac { 5 } { 6 } - i \frac { \sqrt { 11 } } { 6 }\)
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If \( y = f ( g ( h ( x ) ) ) , \) then the derivative \( y ^ { \prime } \) is given by which for
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