What is a Quadratic Equation?
A quadratic equation in the variable x is an equation of the form:
If a = 0, the term ax² becomes 0, reducing the equation to bx + c = 0, which is a linear equation, not a quadratic equation!
Writing Equations in Standard Form
Any quadratic equation should be rearranged with terms in descending order of powers of x:
Check if (x - 2)² + 1 = 2x - 3 is a Quadratic Equation
Solution:
1. Expand LHS using (a - b)² = a² - 2ab + b²:
x² - 4x + 5 = 2x - 3
2. Rearrange all terms to RHS = 0:
x² - 6x + 8 = 0
Since it is in the form ax² + bx + c = 0 with a = 1 ≠ 0, it IS a quadratic equation.
What is a Root?
A real number α is called a root or zero of the quadratic equation ax² + bx + c = 0 if:
Since the highest power of x is 2 (degree = 2), every quadratic equation has at most 2 real roots.
Key Points to Remember
Standard form of a quadratic equation is ax² + bx + c = 0 where a ≠ 0.
The degree of a quadratic equation is always 2.
A quadratic equation has at most two real roots.