What is the special cases of quadratic equation?

Answer 1

The standard form of a quadratic equation is: ax^2 + bx + c = 0. Depending on the values of the constants (a, b, and c), a quadratic equation may have 2 real roots, one double roots, or no real roots.

There are many "special cases" of quadratic equations.

1. When a = 1, the equation is in the form: x^2 + bx + c = 0. Solving it becomes solving a popular puzzle: find 2 numbers knowing their sum (-b) and their product (c). If you use the new Diagonal Sum Method (Amazon e-book 2010), solving is fast and simple.

Example: Solve x^2 + 33x - 108 = 0.

Solution. Roots have opposite signs. Write factor pairs of c = -108. They are: (-1, 108),(-2, 54),(-3, 36)...This sum is -3 + 36 = 33 = -b. The 2 real roots are -3 and 36. There is no needs for factoring.

2. Tips for solving 2 special cases of quadratic equations.

a. When a + b + c = 0, one real root is (1) and the other is (c/a).

Example: the equation 5x^2 - 7x + 2 = 0 has 2 real roots: 1 and 2/5

b. When a - b + c = 0, one real roots is (-1) and the other is (-c/a)

Example: the equation 6x^2 - 3x - 9 = 0 has 2 real roots: (-1) and (9/6).

3. Quadratic equations that can be factored.

The standard form of a quadratic equation is ax^2 + bx + c = 0. When the Discriminant D = b^2 - 4ac is a perfect square, this equation can be factored into 2 binomials in x: (mx + n)(px + q)= 0. Solving the quadratic equation results in solving these 2 binomials for x. Students should master how to use this factoring method instead of boringly using the quadratic formula.

When a given quadratic equation can be factored, there are 2 best solving methods to choose:

a. The "factoring ac method" (You Tube) that determines the values of the constants m, n, p, and q of the 2 above mentioned binomials in x.

b. The Diagonal Sum Method (Amazon ebook 2010) that directly obtains the 2 real roots without factoring. It is also considered as "The c/a method", or the shortcut of the factoring method. See the article titled" Solving quadratic equations by the Diagonal Sum Method" on this website.

4. Quadratic equations that have 2 roots in the form of 2 complex numbers.

When the Discriminant D = b^2 - 4ac < 0, there are 2 roots in the form of 2 complex numbers.

5. Some special forms of quadratic equations:

- quadratic equations with parameters: x^2 + mx - 7 + 0 (m is a parameter)

- bi-quadratic equations: x^4 - 5x^2 + 4 = 0

- equations with rational expression: (ax + b)/(cx + d) = (ex + f)

- equations with radical expressions.