Karteikarten
Quadratic Equation
Karten in diesem Stapel (22)
- 1Quadratic EquationAny equation of the form:-ax^2+bx+c=0 is quadratic equation.If a≠0
a,b,c∈R
Equation:-It involves L.H.S and R.H.S
ax^2+bx+c=0
Expression:-
ax^2+bx+c
Identity
It is an equation which is valid for all 'x'.
For eg
(a+b)^2=a^2+b^2+2ab
It is valid for all a and b.
Every quadratic equation has exactly 2 roots.(could be real or complex)
If any equation looking like quadratic equation has more than 2 roots then it is an identity. - 2Sum of zeros and product of zerosalpha+beta=
-b/a
alpha X beta =c/a - 3Quadratic Formula
- 4Inequalities(x-3)*(x-4)<0
Then 30
x>4 and x<3
(x-4)/(x-3)<0
Hence, 3 - 5Wavy Curve MethodIn this method,inequalities in equation are solved by making making curves like this in diagram.
- 6Question related to wavy curve methodIn this all entities are converted into factor and the factor having odd power is kept but even is removed.
- 7Some important resultsFor ax^2+ b x + c where a≠0
Here,if D<0 and a>0 this for all real values of x. ax^2+bx+c>0
Also,for ax^2+bx+c D<0 and a<0
here for all real x ax^2+bx+c<0 - 8PrecautionsTake care of negatives.
Take care while determining values which are not to be taken and exclude them.
Take care of expression x*(4^x-1) as it is always positive.
Also,x^2+4 like entities are always positive.
When equality is only less than or greater than then numerator can't be zero.
Verify value of x obtained. - 9Some important results1.If D<0 then roots of quadratic equation are imaginary.(both)
2.If D>0 the roots of the equation are real.(both)
3.If D=0 the roots of the equation are equal and real.
4.If D is perfect square then roots are rational.
5.If D is not a perfect square the roots are irrational and conjugate of each other.
Conjugate-It is a number used for rationalisation. - 10For general eqautions1.Irrational roots exist in pairs and are conjugate of each other.
2.Rational roots may exist in pairs.
3.Equation of n degree has maximum n
real roots out total of n roots.
4.Imaginary roots always exist in pairs and conjugate of each other.
Odd degree equation always has atleast 1 real roots and others can be imaginary.
Even degree equation can have all imaginary roots. - 11Conditions for integral roots1.D is a perfect square
2.a=1
3.b,c∈Z
Questions related to this are frequently asked in I.I.T's. - 12For cubic equationFor ax^3+bx^2+c*x+d=0
α+β+γ=-b/a
αβ+βα+γβ=c/a
α*β*γ=-d/a - 13For bi-quadratic equationax^4+bx^3+c*x^2+d*x+e=0
α+β+γ+δ=-b/a=∑α(Odd,Negative)
αδ+αγ+αβ+βγ+βδ+γδ=c/a=∑α*β(Even,Positive)
αβγ+αβδ+αδγ+βδγ=-d/a
αβδγ=e/a
Coefficient chosen depends on distance of coefficient from x^4 coefficient. - 14For n degree equationa 0*x^n+a 1*x^n-1+...........+a n=0
Note There are n+1 terms
Constant term and all coefficients belongs to real numbers
Suppose x1,x2,x3,x4,...... are roots of equation.
1.∑x1=-a1/a0
2.∑x1*x2=a2/a0
3.∑x1*x2*x3=-a3/a0
4. x1*x2*x3*x4......x n=(-1)^n*an/a0 - 15Cubic equationIn cubic ax^3+bx^2+c*x+d=0 if α,β,γ are roots
Then,ax^3+bx^2+c*x+d=a(x-α)(x-β)(x-γ) - 16Transformation of rootsIn we make equation in x from roots α,β only if roots are symmetric.
Suppose ax^2+bx+c=0
Then,if α,β are roots then,we take aα^2+bα+c=0 is taken relationship between x and α is taken and equation is found - 17Graphs of quadratic equation
- 18Reasons for these graphsD>0 then roots are real and distinct
D=0 the roots are real and equal(same/repeated)
D<0 then roots are imaginary
In ax^2+bx+c=0 if x=∞ then f(x)=+∞ if a>0 and if a<0 then f(x)=-∞ - 19Vertex and axis of parabolaAxis-It is the line about which parabola is symmetrical.Vertex lies on axis of parabola.
y=ax^2+bx+c has axis parallel to y axis.
x=a*y^2+by+c has axis parallel to x axis.
We can know the quadratic equation if we know atleast 3 points through which it passes and by knowing the axis which is parallel to it,s axis.
- 20Vertex,s CoordinatesIt is (-b/2a,-D/4a)
For ax^2+bx+c=0 a>0 has y min=-D/4a
For a<0 y max=-D/4a
This can be derived by complete square method.
Note y=x^2+a for y min,x=0 - 21To find a variable coefficient of biquadratic equationConvert biquadratic into quadratic equation find variable by satisfying conditions of t=ax^2+bx+c and equation obtained that will give variable's value.
- 22Some results on iotai=√-1
-i^2=1
i^2=-1
1/i=-i
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- Karteikarten am Quadratic Equation, erstellt von Prajjwal Sanskar am 16/04/2018.
- Erstellt von
- Prajjwal Sanskar
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