In Exercises 127–130, solve each equation by the method of your choice. √2 x^2 + 3x - 2√2 = 0

Verified step by step guidance

1. First, recognize that the given equation \( \sqrt{2} x^2 + 3x - 2\sqrt{2} = 0 \) is a quadratic equation in the form \( ax^2 + bx + c = 0 \).
2. Identify the coefficients: \( a = \sqrt{2} \), \( b = 3 \), and \( c = -2\sqrt{2} \).
3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \).
4. Substitute the identified coefficients into the quadratic formula: \( x = \frac{-3 \pm \sqrt{3^2 - 4(\sqrt{2})(-2\sqrt{2})}}{2(\sqrt{2})} \).
5. Simplify the expression under the square root and the entire formula to find the values of \( x \).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a systematic way to find the roots of any quadratic equation. The term under the square root, known as the discriminant (b² - 4ac), determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots. Mastery of this formula is crucial for solving quadratic equations.

Radical Expressions

Radical expressions involve roots, such as square roots, and can complicate the solving process of equations. In the given equation, √2 x^2 indicates the presence of a square root, which may require isolating the radical and squaring both sides to eliminate it. Understanding how to manipulate radical expressions is vital for solving equations that include them.