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A polynomial of degree 2 (i.e.ax2 + bx + c) is called a quadratic polynomial where a ≠ 0 and a, b, c are real numbers.
The general form of quadratic equation is ax2 + bx + c = 0, where a,b,c are real numbers and a ≠ 0 Since a ≠ 0, quadratic equations, in general are of the following types :-
(i) b = 0, c ≠ 0i.e., of the type ax2 + c = 0.
(ii) b ≠ 0c = 0, i.e. of the type ax2 + bx = 0.
(iii) b = 0, c = 0, i.e. of the type ax2 = 0.
(iv) b ≠ 0, c ≠ 0i.e., of the type ax2 + bx + c = 0.
It is classified into two categories:
(i) Pure quadratic equation (of the form ax2 + c = 0 i.e., b = 0 in ax2 + bx + c)
e.g. x2 - 4 = 0 and 3x2 + 1 = 0 are pure quadratic equations.
(ii) Affected quadratic equation (of the form ax2 + bx + c = 0, b ≠ 0).
e.g. x2- 2x - 8 = 0 and 5x2 + 3x - 2 = 0 are affected quadratic equations.
For a quadratic polynomial p(x) = ax2 + bx + c, those values of x for which ax2 + bx + c = 0 is satisfied, are called zeros of quadratic polynomial p(x), i.e. if p(α) = aα2 + bα + c = 0 then α is called the zero of quadratic polynomial.
The value of x which satisfies the given quadratic equation is known as its root. The roots of the given equation are known as its solution.
General form of a quadratic equation is :
ax2 + bx + c = 0
or 4a2x2 + 4abx + 4ac = 0 [Multiplying by 4a]
or 4a2x2 + 4abx = –4ac [By adding b2 both sides]
or 4a2x2 + 4abc + b2 = b2 – 4ac
or (2ax + b)2 = b2 – 4ac
Taking square root of both the sides
2ax + b =
Hence, roots of the quadratic equation ax2 + bx + c = 0 are
A quadratic equation is satisfied by exactly two values of ‘a’ which may be real or imaginary. The equation, ax2 + bx + c = 0 is :
Consider the quadratic equation, ax2 + bx + c = 0 having αβ as its roots and b2 – 4ac is called discriminate of roots of quadratic equation. It is denoted by D or Δ.
Roots of the given quadratic equation may be
(i) Real and unequal (ii) Real and equal (iii) Imaginary and unequal.
Let the roots of the quadratic equation ax2 + bx + c = 0 (where a ≠ 0,b,c ∈ R ) be α and β then
The nature of roots depends upon the value of expression ‘b2 - 4ac’ within the square root sign. This is known as discriminate of the given quadratic equation.
Consider the Following Cases:
Case-1 When b2 - 4ac > 0, (D > 0)
In this case roots of the given equation are real and distinct and are as follows
(i) When a ≠ 0,b,c ∈ Q and b2 - 4ac is a perfect square
In this case both the roots are rational and distinct.
(ii) When a ≠ 0,b,c ∈ Q and b2 - 4ac is not a perfect square
In this case both the roots are irrational and distinct. [See remarks also]
Case-2 When b2 - 4ac = 0, (D = 0)
In this case both the roots are real and equal to -b/2a.
Case-3 When b2 - 4ac < 0, (D < 0)
In this case b2 – 4ac < 0, then 4ac – b2 > 0
i.e. in this case both the root are imaginary and distinct.
question 1. Check whether the following equations are quadratic or not.
Solution: (i) can be rewritten as
It is of the form ax2 + bx + c = 0
Therefore, the given equation is a quadratic equation.
(ii) As and (x + 2) (x - 2) = x2 - 4
Therefore, we have x2 + x + 8 = x2 - 4
⇒ x + 12 = 0
It is not of the form ax2 + bx + c = 0
Therefore, the given equation is not a quadratic equation.
⇒ x + x-1 - 2 = 0
Since x + x-1 - 2 is not a quadratic polynomial.
∴ is not a quadratic equation
question 2. In each of the following, determine the value of k for which the given value is a solution of the equation.
Solution: (i) …(i)
Since x = 2 is a solution of the equation
Now substituting x = 2 in equation (i), we get
(ii) Since x = –2 is a solution of the equation
Now substituting x = – 2 in (i), we get
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