What is a quadratic equation? A Quadratic equation is a statement ...

Pre-calculus: Quadratic Roots
What is a quadratic equation?
A Quadratic equation is a statement for equating a polynomial of degree 2
to zero:
P(x) = 0
deg[P(x)] = 2
Examples:
x 2 + 2x + 1 = 0
3x 2 + 1− 5 = 0
What is a root of a quadratic equation?
A number, α is a root of a quadratic equation if and only if P ( α ) = 0.
What is the difference between the term ‘root’ and ‘zero’?
The term ‘root’ is for equations while the term ‘zero’ is for a polynomial.
We say α is a root of a polynomial equation P ( x ) = 0.
We say α is a zero of the polynomial P ( x ) such that P ( α ) = 0.
How many roots does a quadratic equation has?
According to the fundamental theorem of algebra, a quadratic equation has
at most 2 roots (including real and imaginary roots).
What is the general form of a quadratic equation?
A x 2 + Bx + C = 0.
What is the determinant of a quadratic equation? What does it tell?
The determinant of a quadratic equation is D = B 2 - 4 A C. It tells the nature
of the roots:
determinant \ nature number of roots nature of roots
exactly zero 1 real, double
greater than zero 2 both are real
less than zero 2 both are imaginary

Pre-calculus: Quadratic Roots
Why the determinant of a quadratic equation tells the nature of the roots?
Since the general form of a quadratic equation,
A x 2 + B x + C = 0
Can be represented as
A { x + [ B / (2 A) ] } 2 + [ C - B 2 / ( 4 A ) ].
The term C - B 2 / ( 4 A ) is the y-coordinate of the vertex point, which
happens to be - D / ( 4 A ).
Therefore if D = 0, the root of the quadratic equation is exactly the vertex.
If D > 0 and A > 0, that means the minimum point is below x-axis and the
parabola is opened upwards, then there must be two real roots.
Similarly, If D > 0 and A < 0, that means the maximum point is above x-axis
and the parabola is opened downwards, then there must be two real roots.
For D < 0 and A > 0, that means the minimum point is above x-axis and the
parabola is opened upwards, then there must not be any real roots.
For D < 0 and A < 0, that means the maximum point is below x-axis and the
parabola is opened downwards, then there must not be any real roots.
What is the quadratic formula?
The quadratic formula is a formula to compute the roots of
A x 2 + B x + C = 0.
The roots of the quadratic equations can be founded by
x = −B ± B2 − 4AC
2A
What are the properties of product of the roots and the sum of the roots of
a quadratic equation?
The product of the roots is C / A and the sum of the roots is - B / A.
To see these, if we represent a quadratic equation with its roots α1, α2 in
this way:
A(x − α 1 )(x − α 2 ) = 0
After expansion,
Ax 2 − A(α 1 + α 2 ) + A(α 1α 2 ) = 0
By comparing the coefficient with A x 2 + B x + C = 0, we get
B = - A * (sum of roots)"" " and C = A * (product of roots)
which yields
sum of roots = - B / A" " " and product of roots = C / A