The standard form of a quadratic equation is: ax2+bx+c = 0 where a, b and c are known values and a ≠0. Also, x is a real variable.
If the value of a is 0, then the equation will become a linear equation.
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:
- When we have a positive value of b2- 4ac, then we will get two Real solutions
- When the value of b2- 4ac is zero, then we will get only one Real solution
- 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
Let the roots of the equation ax2+ bx + c be α and β.
- Then, the sum of the roots: α + β = -
- The product of the roots = α β =
- 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