In each of the following, solve the given quadratic equation...

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.
$$1 / 2 x^{2}+5 x+2=0$$

Key Concepts

Approximation of Irrational Solutions

When the quadratic formula yields irrational numbers due to a non-perfect square discriminant, the solutions can be expressed exactly using radicals. However, for practical purposes, these irrational values are often approximated to a desired degree of accuracy, such as to the nearest thousandth, using a calculator.

Simplifying Radical Expressions

Simplifying expressions, especially those involving radicals, is a critical step when applying the quadratic formula to ensure that the solutions are expressed in their simplest exact form. This process may involve factoring out perfect squares from under the radical or rationalizing denominators.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), provides a systematic method for finding the roots or solutions of any quadratic equation. It is derived by completing the square on the general quadratic equation and is essential because it applies universally regardless of the specific values of a, b, and c.

Discriminant

The discriminant, given by b² - 4ac, is the expression under the square root in the quadratic formula. It determines the nature and number of the roots: if it is positive, the quadratic equation has two distinct real roots; if it is zero, there is exactly one real root (a repeated solution); and if it is negative, the equation has two complex conjugate roots.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in one variable, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants and a is nonzero. It is a fundamental concept in algebra and is used to model various real-world phenomena as well as in higher-level mathematics.

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