Which equation has the solutions [tex]x=\frac{-3 \pm \sqrt{3} i}{2}[/tex]?
A. [tex]x^2+3 x+3=0[/tex]
B. [tex]2 x^2+6 x+9=0[/tex]
C. [tex]2 x^2+6 x+3=0[/tex]
D. [tex]x^2+3 x+12=0[/tex]
Answer (1)
Find the sum of the roots: x 1 + x 2 = − 3 .
Find the product of the roots: x 1 x 2 = 3 .
Substitute the sum and product into the quadratic equation form: x 2 − ( x 1 + x 2 ) x + x 1 x 2 = 0 .
Simplify the equation to get the final answer: x 2 + 3 x + 3 = 0 .
Explanation
Problem Analysis We are given the solutions x = 2 − 3 ± 3 i to a quadratic equation and need to find the equation itself.
Setting up the Quadratic Equation Let the solutions be x 1 = 2 − 3 + 3 i and x 2 = 2 − 3 − 3 i . A quadratic equation with roots x 1 and x 2 can be written as ( x − x 1 ) ( x − x 2 ) = 0 . Expanding this, we get x 2 − ( x 1 + x 2 ) x + x 1 x 2 = 0 . Therefore, we need to find the sum and product of the roots.
Calculating Sum and Product of Roots The sum of the roots is: x 1 + x 2 = 2 − 3 + 3 i + 2 − 3 − 3 i = 2 − 3 + 3 i − 3 − 3 i = 2 − 6 = − 3 The product of the roots is: x 1 x 2 = ( 2 − 3 + 3 i ) ( 2 − 3 − 3 i ) = 4 ( − 3 ) 2 − ( 3 i ) 2 = 4 9 − ( − 3 ) = 4 12 = 3
Forming the Quadratic Equation Substituting the sum and product of the roots into the quadratic equation, we get: x 2 − ( − 3 ) x + 3 = 0 Simplifying the equation: x 2 + 3 x + 3 = 0
Final Answer Comparing the obtained equation with the given options, we find that the correct equation is x 2 + 3 x + 3 = 0 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when designing a parabolic reflector for a solar oven, the equation helps determine the optimal shape to focus sunlight onto a single point, maximizing heat concentration. Similarly, in finance, quadratic equations can model investment returns, helping investors understand potential profits and risks associated with different strategies. These applications demonstrate how mastering quadratic equations provides a foundation for solving real-world problems in diverse disciplines.
