Short notes of Quadratic Equation for IIT – JEE main & advance

Quadratic Equation is the most important chapter from a IIT -JEE main & advance point of view.  So repeat these to score the most!

 

The general form of a quadratic equation in x is, ax2+ bx+ c = 0 , where a , b, c ∈ R & a ≠ 0.

RESULTS :

1. The solution of the quadratic equation , ax² + bx + c = 0 is given by x = -b ± √b2 – 4ac / 2a

The expression b2 – 4ac= D is called the discriminant of the quadratic equation.

2. If α & β are the roots of the quadratic equation ax² + bx + c = 0, then;

(i)α +  β  = – b/a

(ii)  α β= c/a

(iii) α –  β = √D / a .

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3. NATURE OF ROOTS:

(A) Consider the quadratic equation ax²+ bx + c = 0 where a, b, c ∈ R & a ≠ 0 then ;

(i) D > 0 <-> roots are real & distinct (unequal).

(ii) D = 0 <-> roots are real & coincident (equal).

(iii) D < 0 <-> roots are imaginary .

(iv) If p + i q is one root of a quadratic equation, then the other must be the conjugate p –  i q & vice versa. (p, q ∈ R & i = √1).

 

(B) Consider the quadratic equation ax2+ bx + c = 0 where a, b, c ∈ Q & a ≠ 0 then;

(i) If D > 0 & is a perfect square , then roots are rational & unequal.

(ii) If  α = p + q is one root in this case, (where p is rational & √q is a surd) then the other root must be the conjugate of it i.e. β = p – q & vice versa.

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4. A quadratic equation whose roots are α  & β is  (x – α )(x – β) = 0 i.e.

x2– (α+β )x + αβ = 0 i.e. x2 -(sum of roots)x + product of roots = 0.

 

5. Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity.

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6. Consider the quadratic expression , y = ax²+ bx + c , a  ≠ 0 & a , b, c ∈ R then ;

(i) The graph between x , y is always a parabola . If a > 0 then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.

(i)  y > 0 only if a > 0 & b² –  4ac < 0

(ii)  y < 0 only if a < 0 & b² – 4ac < 0

 

7. SOLUTION OF QUADRATIC INEQUALITIES:

ax2 + bx + c > 0 (a ≠ 0).

(i) If D > 0, then the equation ax2+ bx + c = 0 has two different roots x1  < x2

Then

a  > 0  ⇒ X ∈  ( -∞, x1) ∪ (x2, ∞)

a < 0 ⇒ X ∈ (X1, X2)

(ii) If D = 0, then roots are equal, i.e. x1 = x2 .

In that case

a > 0 ⇒  X ∈  ( -∞, x1) ∪ (x1, ∞)

a < 0 ⇒   X ∈ Φ

(iii) Inequalities of the form P(x) / Q(x)  0 can be quickly solved using the method of intervals

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8. MAXIMUM & MINIMUM VALUE

of y = ax²+ bx + c occurs at x = – (b/2a) according as ;

a < 0 or a > 0 . y ∈ [ 4 ac – b2 / 4a , ∞) if a > 0 & y ∈ (- ∞, -4ac – b2 / 4a ] if a < 0

 

9.COMMON ROOTS OF 2 QUADRATIC EQUATIONS [ONLY ONE COMMON ROOT] :

Let α  be the common root of ax² + bx + c = 0 &  a’x2+ b’x + c’ = 0 . Therefore

a α²+ bα+ c = 0 ; a’α² + b’α+ c’ = 0.  By Cramer’s Rule   α2/ bc’ – b’c = α /a’c – ac’ = 1/ ab’- a’b

Therefore, α= ca’- c’a / ab’- a’b = bc’- b’c/ a’c – ac’ .

So the condition for a common root is (ca’ – c’a)² = (ab’ a’b)(bc’ –  b’c)

10. The condition that a quadratic function

f(x , y) = ax²+ 2hxy + by² + 2gx + 2 fy + c may be resolved into two linear factors is that ;

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11. THEORY OF EQUATIONS :

If α1 , α2 , α3 , ……αn are the roots of the equation;

f(x) = a0xn  +  a1xn-1  +  a2xn-2  + …. +  an-1x  +  a n = 0            where              a0 , a1 , …. a n are all real & a 0  ≠ 0

then,

Σ α1 = -a1 / a0 , Σα1 α2 = + a2 / a, Σ α1 α2 α3 = a3 / a0 , …… ,  α1 α2 α3 …….  αn = (-1)n an / a0

Note :

(i) If α is a root of the equation f(x) = 0, then the polynomial f(x) is exactly divisible by  (x – α ) or (x – α) is a factor of f(x) and conversely .

(ii) Every equation of nth degree (n ≥ 1) has exactly n roots & if the equation has more than n roots, it is an identity.

(iii) If the coefficients of the equation f(x) = 0 are all real and  a  + ib is its root, then a  –  ib is also a root. i.e. imaginary roots occur in conjugate pairs.

(iv) If the coefficients in the equation are all rational & a   + √β is one of its roots, then a   – √β is also a root where a  , a ,β ∈ Q & β  is not a perfect square.

(v) If there be any two real numbers ‘a’ & ‘b’ such that f(a) & f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’ .

(vi) Every equation f(x) = 0 of degree odd has at least one real root of a sign opposite to that of its last term.

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12. LOCATION OF ROOTS :

Let f(x) = ax2+ bx + c, where a > 0 & a, b, c ∈  R.

(i) Conditions for both the roots of f (x) = 0 to be greater than a specified number ‘d’ are b2  4ac  ≥  0; f(d) > 0 & (-b/2a) > d.

(ii) Conditions for both roots of f(x) = 0 to lie on either side of the number ‘d’ (in other words the number ‘d’ lies between the roots of f(x) = 0) is f(d) < 0.

(iii) Conditions for exactly one root of f(x) = 0 to lie in the interval (d, e) i.e. d < x < e are b2–  4ac > 0 & f(d) .f(e) < 0.

(iv) Conditions that both roots of f (x) = 0 to be confined between the numbers p & q are (p < q). b2- 4ac ≥  0; f(p) > 0; f(q) > 0 & p < (- b/2a) < q.

 

13. LOGARITHMIC INEQUALITIES

(i) For a > 1 the inequality 0  <  x  <  y  &  logaX  <  logay are equivalent.

(ii) For 0 < a < 1 the inequality 0  <  x  <  y  &  logaX  >  logay are equivalent.

(iii) If a > 1 then logax  < p    ⇒    0 < x < ap

(iv) If a > 1 then logax  > p    ⇒     x > ap

(v) If 0 < a < 1 then logax  < p     ⇒      x > ap

(vi) If 0 < a < 1 then logax  > p     ⇒      0 < x < ap

 

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