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

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Key Concepts

Quadratic Equations

Quadratic equations are second-degree polynomial equations that typically take the form ax² + bx + c = 0. They have two solutions that may be real or complex, and various methods such as factoring, completing the square, or using the quadratic formula can be applied to find these solutions.

Standard Form

Writing a quadratic equation in standard form, ax² + bx + c = 0, is essential as it clearly identifies the coefficients and sets up the equation for further analysis or solution techniques such as completing the square.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting them as a perfect square trinomial plus or minus a constant. This process involves factoring out the coefficient of the x² term if necessary, then finding and adding the square of half the coefficient of x, which simplifies the equation to a form that can be easily solved.

Complex Numbers

Complex numbers are used to express the solutions of quadratic equations when the discriminant (the part under the square root in the quadratic formula) is negative. These solutions are written in the form a ± bi, where i represents the imaginary unit defined by i² = -1, thus extending the set of solutions beyond the real numbers.

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