Logs, Equations and Inequalities

Logarithms and Related Properties

Quadratic Equation An equation of the form ax2 + bx + c = 0, where a(≠ 0), b and c are real numbers is called quadratic equation. The numbers a, b, c are called the coefficients of the quadratic equations.  

Roots of Quadratic Equation

The values of the variable satisfying a given quadratic equation are called its roots. A quadratic equation has exactly two roots which can be real or imaginary.   The roots of the quadratic equation ax2 + bx + c = 0 are given by x=-b±b2-4ac2a. If α is a root of the quadratic equation, then aα2+bα+c=0. If a + b + c = 0, then the quadratic equation ax2 + bx + c = 0 has a root x = 1.  

Relationship Between the Roots and Coefficients of a Quadratic Equation

Let α and β be the roots of the equation ax2 + bx + c = 0, then

Sum of the roots, α+β=-Coefficient of xCoefficient of x2=-ba

Product of the roots, αβ=Constant termCoefficient of x2=ca

A quadratic equation, whose roots are α and β can be written as x-αx-β=0, i.e. ax2+bx+c≡ax-αx-β.  

Nature of Roots of Quadratic Equation

The expression D defined by D=b2-4ac is called discriminant of the quadratic equation ax2 + bx + c = 0, where a(≠ 0), b and c are real numbers.

D > 0, only if the roots of the quadratic equation are real and distinct.

D = 0, only if the roots of the quadratic equation are equal.

D < 0, only if the roots of the quadratic equation are imaginary.  

Some Basic Results

If α=u-v is an irrational root of the quadratic equation ax2 + bx + c = 0, then β=u+v is also a root of the quadratic equation, provided that all the coefficients are rational. If α=u+iv is one of the roots of ax2 + bx + c = 0, then the other root will be β=u-iv, where i =-1. The quadratic equation ax2 + bx + c = 0 has rational roots, if D is a perfect square and a, b, c are rational. If ax2 + bx + c = 0 is satisfied by more than two values of x, then it is an identity and a = b = c.   Condition for Common Root

Let a1x2+b1x+c1=0 and a2x2+b2x+c2=0 be two quadratic equations. Let α be the common root of both the equations. Then

a1α2+b1α+c1=0            .....(1)

a2α2+b2α+c2=0            .....(2)

Solving (1) and (2) by Cramer's rule, we get

α2b1c2-b2c1=αa2c1-a1c2=1a1b2-a2b1

⇒α2=b1c2-b2c1a1b2-a2b1 and  α=a2c1-a1c2a1b2-a2b1

⇒b1c2-b2c1a1b2-a2b1=a2c1-a1c2a1b2-a2b12

⇒a2c1-a1c22=b1c2-b2c1a1b2-a2b1, which is the required condition for the two equations to have a common root. Note:

If two quadratic equations have both roots common, then the coefficients of the corresponding powers of x will have proportional values. If two quadratic equations with rational coefficients have a common irrational root u+v, then both roots will be common. If α is the repeated root of the quadratic equation f(x) = ax2 + bx + c = 0, then α is also the root of the equation f '(x) = 0.  

Relation Between the Roots of a Polynomial Equation of Degree n

Let α1, α2, α3, ..., αn, where a1, a2, a3, ..., an∈R be the roots of the equation fx=anxn+ αn-1xn-1+αn-2xn-2+...+α2x2+α1x+α0=0, an≠0         .....(1)

Then (1) can be written as

anxn+ αn-1xn-1+αn-2xn-2+...+α2x2+α1x+α0=anx-α1x-α2x-α3...x-αn                          .....(2)

Now, on comparing the coefficients of like powers of x in (2), we get

∑α1=-an-1an

∑α1α2=an-2an

∑α1α2α3=-an-3an

.  .  .  .  .  .  .  .  .  .  .  .  .  .  .

.  .  .  .  .  .  .  .  .  .  .  .  .  .  .

α1α2α3...αn=-1na0an  

If α, β and γ are the roots of the cubic equation ax3+bx2+cx+d=0, then

 α+β+γ=-ba

 αβ+βγ +γα =ca

 αβγ=-da   

Some Important Results

Let fx=anxn+ αn-1xn-1+αn-2xn-2+...+α2x2+α1x+α0​ be a polynomial having degree n.

A polynomial equation fx=0 of degree n has n roots, which may be real or imaginary. If all the coefficients a1, a2, a3, ..., an∈R , then the complex roots occur in conjugate pairs. This means that the number of complex roots is always even. If the degree of the polynomial equation, n, is odd, the number of real roots will also be odd. Thus, at least one of the roots will …