Solve each equation using the most efficient method:...

Solve each equation using the most efficient method:
factoring, square root property of equality, or the quadratic formula. Write your answer in both exact and approximate form (rounded to hundredths). Check one of the exact solutions in the original equation.
$$2 p^{2}-4 p+11=0$$

Key Concepts

Solution Verification

Verifying solutions involves substituting the found values back into the original equation to check their validity. This step is crucial in ensuring that the solutions are correct, helping to catch any errors made during the solving process.

Exact and Approximate Solutions

Exact solutions are expressed in precise mathematical terms (often involving radicals or complex numbers), while approximate solutions provide numerical estimates usually rounded to a certain number of decimal places. Understanding both forms is important for precise mathematical communication and practical applications.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ? 0. Solving quadratic equations is a fundamental concept in algebra, and understanding their structure helps in determining the best method for finding their solutions.

Quadratic Formula

The quadratic formula, given by x = (-b ± ?(b² - 4ac)) / (2a), is a universal method for solving any quadratic equation. It provides the solutions by directly substituting the coefficients, and it is especially useful when other methods like factoring or applying the square root property are not applicable.

Discriminant

The discriminant, which is the part of the quadratic formula under the square root (b² - 4ac), indicates the nature of the solutions of a quadratic equation. A positive discriminant suggests two distinct real solutions, zero indicates one repeated real solution, and a negative discriminant means the solutions are complex conjugates.

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