Solve each equation by completing the square. Write complex...

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Key Concepts

Complex Numbers

Complex numbers extend the real number system to include numbers of the form a ± bi, where a and b are real numbers and i represents the square root of -1. This concept is crucial for solving quadratic equations with negative discriminants, as it allows for meaningful solutions when no real roots exist.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial plus a constant. This transformation simplifies solving the equation by isolating the squared term, thereby enabling the extraction of the variable through the square root process. It is particularly useful when the quadratic does not factor neatly.

Quadratic Equations

Quadratic equations are second-degree polynomial equations typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. They may have two solutions that can be real or complex, and various techniques such as factoring, the quadratic formula, or completing the square are applied to find these solutions.

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