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, ax ^{2}+ 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 ± √*b ^{2} – 4ac */ 2a

The expression b^{2} – 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 .

**ALSO READ : **

Short Notes of Complex Numbers for IIT-JEE mains and advance

**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.

**ALSO READ:**

Important Questions Of Complex Numbers for IIT-JEE mains and advance

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

x

^{2}– (α+β )x + αβ = 0 i.e. x^{2}-(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.

**MUST READ :-**

**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:**

ax^{2} + bx + c > 0 (a ≠ 0).

**(i)** If D > 0, then the equation ax2+ bx + c = 0 has two different roots x1 < x_{2}

Then

a > 0 ⇒ X ∈ ( -∞, x_{1}) ∪ (x_{2}, ∞)

a < 0 ⇒ X ∈ (X_{1}, X_{2})

**(ii)** If D = 0, then roots are equal, i.e. x_{1} = x_{2} .

In that case

a > 0 ⇒ X ∈ ( -∞, x_{1}) ∪ (x_{1}, ∞)

a < 0 ⇒ X ∈ Φ

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

**ALSO READ :-**

**8. MAXIMUM & MINIMUM VALUE **

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

a < 0 or a > 0 . y ∈ [ 4 ac – b^{2} / 4a , ∞) if a > 0 & y ∈ (- ∞, -4ac – b^{2} / 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 α_{1} , α_{2} , α_{3} , ……α_{n} are the roots of the equation;

f(x) = a_{0}x^{n} + a_{1}x^{n-1 } + a_{2}x^{n-2 } + …. + a_{n-1}x + a n = 0 where a0 , a1 , …. a n are all real & a 0 ≠ 0

then,

Σ α

_{1}= -a_{1}/ a_{0}, Σα_{1}α_{2}= + a_{2}/ a_{0 }, Σ α_{1}α_{2}α_{3}= a_{3}/ a0 , …… , α_{1}α_{2}α_{3}……. αn = (-1)^{n}a_{n}/ a_{0}

**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 + i*b* is its root, then a – i*b* 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.

**ALSO READ :-**

**12. LOCATION OF ROOTS :**

Let f(x) = ax^{2}+ 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 b^{2} 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 b^{2}– 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 & log_{aX} < log_{a}y are equivalent.

**(ii)** For 0 < a < 1 the inequality 0 < x < y & log_{aX } > log_{a}y are equivalent.

** (iii)** If a > 1 then log_{a}x < p ⇒ 0 < x < a^{p}

**(iv)** If a > 1 then log_{a}x > p ⇒ x > a^{p}

** (v)** If 0 < a < 1 then log_{a}x < p ⇒ x > a^{p}

**(vi)** If 0 < a < 1 then log_{a}x > p ⇒ 0 < x < a^{p}

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