Quick Review: Quadratic Equations

Roots of a Quadratic Equation

• A root of the equation f(x) = 0 is when the value of x makes f(x) = 0. We can say that x = a is the root of f(x) = 0. Here, f(a) is the value of the polynomial f(x) at x = 0 and is obtained by replacing x by a in f(x).
• If there is a quadratic equation ax² + bx + c = 0, then the roots of this equation will be:
• In the above mentioned formula, b²- 4ac is the discriminant as it can discriminate between the possible types of solutions:

1. When we have a positive value of b²- 4ac, then we will get two Real solutions
2. When the value of b²- 4ac is zero, then we will get only one Real solution
3. When we have a negative value of b²- 4ac, then we will get two Complex solutions (i.e. the answer will include imaginary numbers)

Relation between Roots and Coefficients

• Then, the sum of the roots: α + β = -
• The product of the roots = α β =

Key Points

• If p+ √q is a root of a quadratic equation, then its other root is p- √q
• When D ≥ 0, then ax²+ bx + c can be expressed as a product of two linear factors.
• If α and βare the roots of ax²+ bx + c, then we can write it as: x² - (α + β)x + α β = 0