1. Quadratic expression
A polynomial of degree two of the ax2 + bx + c, a ¹ 0 is called q quadratic expression in x.
2. Quadratic equation :
An equation ax2 + bx + c = 0 a ¹ , a,b,c, Î R has teo and only two roots given by
a = (-b+Ö(b2 -4ac) / 2a ) and b = (-b-Ö(b2 -4ac)/2a)
3. Nature of roots :
Nature of root of the given equation depends upon the nature of its discriminant D I.e. b2 - 4ac
Suppose a,b,c Î R a¹ 0 then
(i) If D > 0 Þ rootsare real and distinct (unequal)
(ii) If D = 0 Þ roots are real and equal (coincident)
(iii) If D < 0 Þ roots are imaginary and unequal i.e.
non real complex number .
Suppose a,b,c Î Q a ¹ 0 then
(i) If D > 0 and D is a perfect squar Þ roots are rational & unequal
(ii) If D > 0 and D is not a perfect square Þ roots are irrational and unequal
For a quadratic equation their will exist exactly 2 roots real or imaginary If the equation ax2 + bx + c =0 is satisfied for more then 2 distinct values of x, then it will be an identity & will be satisfied by all x. also in this case a = b =c =0
4. Conjugate roots:
Irrational roots and complex roots occur in conjugate pairs i.e
If one root a + Öb , then other roots a - Öb
5. Sum of roots :
S = a + b = (-b/a) = (-Coefficient of x) /( Coefficient x2)
P = ab = (c/a) = (constant term)/ (Coefficient of x2)
Product of roots :
S¹¹ = (c/a) = (constant term) /(Coefficient of x2)
6. Formation of equation with given roots :
X2 - Sx + P =0
Þ x2 - (Sum of roots) x + product of roots =0
7. Roots under particulars cases:
For the quation ax2 + bx + c =0 a ¹ 0
(i)If b =0 Þ roots are of equal magnitude but of opposite sign.
(ii)If c=0 Þ one is zero and other is –b/a
(iii)If b = c = 0Þ both roots are zero
(iv)If a= c Þ roots are reciprocal to each other
(v)If a > 0, c < 0 or a < 0, c > 0 Þ roots are of opposite signs
(vi)If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 Þ both roots are –ve
(vii)If a > 0, b < 0, c < 0 or a < 0, b < 0, c < 0 Þ both roots are + ve
8. Symmetric function of roots:
If roots of quadratic equation ax2 + bx + c, a ¹ 0 are a and b, then
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
9. Condition for common roots:
The equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have
(1) one common roots if 
(2) Both roots common if (a1/a2) = (b1 /b2) = (c1/c2)
10. Maximum and minimum value of quadratic expression:
In a quadratic expression ax2 + bx + c =
Where D = b2 - 4ac
(i) if a > 0, quadratic expression has minimum value  and there is no maximum value.
(ii) If a < 0, quadratic expression has maximum value
and there is no minimum value.
11. Location of roots:
Let f(x) = ax2 + bx +c, a ¹ 0 then w.r.to f(x) = 0
(i) If k lies between the roots then a.f(k) < 0
(necessary & sufficient)
(ii) If between k1 & k2 there is exactly one root of k1, k2 themselves are not roots
F(k1 . f(k2) < 0 (necessary & sufficient)
(iii) If both the roots are less then a number k
D ³ 0, a.f(k) > 0, (-b/2a) < k (necessary & sufficient)
(iv) If both the roots are greater then k
D ³ 0, a.f(k) > 0, (-b/2a) > k (necessary & sufficient)
(v) If both the roots lies iv the interval (k1 ,k2)
D ³ 0, a.f(k1) > 0, a.f(k2) > 0, k1 < (-b/2a) < k2
(vi) If k1, k2 lies between the roots a.f(k1) < 0, a.f(k2) < 0
(vii) l will be the repeated root of f(x) = 0 if f(l) = 0 and f’(l) = 0
12. For cubic equation ax3 + bx2 + cx + d = 0:
We have a+b+g = (-b/a), ab + bg + ga = (c/a) and abg = (-d/a) where a,b,g are its roots.
13. For biquadratic equation ax4 + bx3 + cx2 +dx + e =0 :
We have a+b+g+d = -(b/a), abg + bgd + gda + gdb = (-d/a)
ab + ag + ad + bg + bd + gd = (c/a) and abgd = (e/a)
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