By any method, determine all possible real solutions of each equation. 2 x^2-x-3=0

Name: Problem 20, Chapter 0: Finite Mathematics and Applied Calculus
Uploaded: 2021-08-30T23:50:39-08:00
Duration: 3 min 12 s
Channel: Charles Machakwa
Description: By any method, determine all possible real solutions of each equation. 2 x^2-x-3=0

step 1 Here we have the equation 2x squared minus x minus 3 is equal to 0 and we are to find any possib

By any method, determine all possible real solutions of each...

Key Concepts

Completing the Square

Completing the square is another technique used to solve quadratic equations by rewriting the equation in the form (x + d)² = e, where d and e are expressions involving the original coefficients. This transformation not only helps in solving the equation but also provides insight into the geometric interpretation of quadratics, such as finding the vertex of a parabola.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac))/(2a), provides a general method for solving any quadratic equation regardless of its factorability. This method consistently yields the exact roots of the equation and is particularly useful when other methods, such as factoring, are difficult or not applicable.

Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic as a product of two binomials. Once the equation is factored, the Zero Product Property is applied, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. This method is efficient for quadratics that can be readily factored over the real numbers.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a is not zero. These equations arise frequently in various areas of mathematics and science and require specific methods for finding their solutions, which can include real or complex roots.

Discriminant

The discriminant is the part of the quadratic formula given by b² - 4ac, and it plays a crucial role in determining the nature of the roots of a quadratic equation. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real repeated root, and a negative discriminant indicates a pair of complex conjugate roots.

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