The Quadratic Formula - Revision Notes

In these revision notes for The Quadratic Formula, we cover the following key points:

• What is a quadratic formula? What is it used for?
• What is the discriminant? What does it tell about a quadratic equation?
• What are some applications of quadratic equations in practice?
• What are Vieta's Formulas? Why do we use them?
• How do Vieta's Formulas help us find the original coefficients of a quadratic equation?

The Quadratic Formula Revision Notes

The Quadratic Formula is a standard method for solving quadratic equations. It allows us to find the roots of a quadratic equation without the need to factorize it. This formula is as follows:

Another advantage of the Quadratic Formula is that it allows us to detect immediately when a quadratic equation cannot be solved - a condition that is quite impossible to verify through the other methods. This is because not all quadratic equations have two distinct roots. Some quadratic equations may have a single root or no root at all.

The expression inside the root makes the distinction between various cases of quadratic equations in regard to the number of solutions it has. This part is called discriminant, Δ. Thus, we have

We can identify the following cases in a quadratic equation:

1. If ∆ > 0 (b2 - 4ac > 0), then the corresponding quadratic equation has two distinct roots (solutions):
   x₁ = -b - √∆/2a
   i.e.
   x₁ = - b - √b² - 4ac/2a
   and
   x₂ = -b + √∆/2a
   or
   x₂ = - b + √b² - 4ac/2a
2. If ∆ = 0 (b2 - 4ac = 0), then the corresponding quadratic equation has a single root (more precisely, two equal roots, or solutions):
   x₁ = x₂ = -b/2a
   This is because - √Δ = + √Δ = 0.
3. If ∆ < 0 (b2 - 4ac < 0), then the corresponding quadratic equation has a no roots (no solutions): This is because it is impossible to calculate the square root of a negative number.

Quadratic equations are commonly used in many fields in practice, such as in geometry, natural sciences, etc.

If the roots x¹ and x² of a quadratic equation are known but not the original equation, and we are looking for info about the coefficients a and b and the constant c that allow us determine the original equation, we use two formulas, known as Vieta's Formulas. They are:

1. The roots sum formula:
   x₁ + x₂ = - b/a
2. The roots product formula:
   x₁ ∙ x₂ = c/a