A polynomial of degree 2 is called quadratic polynomial. The name Quadratic comes from "quad" means square,
because the variable gets squared (like x2).
The general form of a quadratic polynomials is
ax2 + bx + c, where a, b, c are real numbers such that a ≠ 0 and x is a
variable. When we equate this polynomial to zero, we get a quadratic equation
i.e. ax2 + bx + c = 0 and when we
write the terms of p(x) in descending order of their degrees, then we get the "Standard
Form" of a Quadratic Equation i.e. ax2 + bx + c = 0. We can also denote a quadratic polynomial ax2 + bx + c by p(x) i.e. p(x) = ax2 + bx + c. It is also called an
"Equation of degree 2"
(because of the power "2" on the x).
real numbers a, b and c are known as the coefficients and are also known as the real
constants because they are fixed and do not depend on the values of
are some more examples:
Example 1: 2x2 -3x + 4 =
0 (Here a=2, b=-3 and c=4)
Example 2: x2 + 9 = 0 (Here
a=1, and we don't usually write "1x2", b=0 and c=9)
Example 3: x2 = 3x – 1
(In this case we will move all terms to left hand side i.e. x2 − 3x
+ 1 = 0. Here a=1, b=−3, c=1)
Example 4: 2(p2 – 2p) =
5 (Solve the brackets and move all terms to left hand side i.e. 2p2
− 4p − 5 = 0. Here a=2, b=−4, c=−5)
Example 5: Jia and Tia together
have 45 pencils. Both of them lost 5 pencils each, and the product of the
number of pencils they now have is 124. Find out how many pencils each had.
Solution: Let the number of pencils
Jia had be x.
Then the number of pencils Tia had = 45 – x
(left out of total belongs to Tia)
The number of pencils left with Jia, when she
lost 5 pencils = x – 5
The number of pencils left with Tia, when she
lost 5 pencils
= 45 – x – 5
= 40 – x
Value of Quadratic
Let’s p(x) = ax2 + bx + c be a quadratic polynomial and let ‘α’ be the real
Then, aα2 + bα + c is known as the value of the quadratic polynomial
p(x) and it is denoted by p (α), i.e., p(α)= aα2 + bα + c.
Thus, the value of a quadratic polynomial ax2 + bx + c at x = α is the
value of the expression ax2 + bx + c obtained by substituting x = α.
Example1: Find the values of the
quadratic polynomial p(x) = 2x2 -3x + 5 at x = -1 and x = 2.
Solution: Here, p(x) = 2x2
-3x + 5.
Case 1: Now substitute the value
of x by -1 in the given quadratic polynomial.
p (-1) = 2 x (-1)2 – 3 x (-1) + 5
= 2 + 3 + 5
Case 2: Now substitute the value
of x by 2 in the given quadratic polynomial.
p (2) = 2 x (2)2 – 3 x (2) + 5
= 8 - 6 + 5
Zero of a Quadratic
number ‘α’ is called a zero of a quadratic polynomial p(x) = ax2 + bx + c, if p (α) = 0.
Thus, a zero of a quadratic polynomial is the value of the variable for which
the value of the polynomial becomes zero. Every quadratic polynomial can have at
most two zeros.
Example 1: Show that 2 is a zero of
the quadratic polynomial p(x) = x2 = x – 6.
Solution: Here, p(x) = x2
= x – 6.
Substitute value of x by 2
Therefore, p (2) = 22 + 2 – 6
= 4 + 2