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 EquationThe 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 ResultsIf 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
Solving (1) and (2) by Cramer's rule, we get
⇒α2=b1c2-b2c1a1b2-a2b1 and α=a2c1-a1c2a1b2-a2b1
⇒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
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
α1α2α3...αn=-1na0anIf α, β and γ are the roots of the cubic equation ax3+bx2+cx+d=0, then
αβ+βγ +γα =ca
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 be real. If fx=x-αrgx, where g (x) is a polynomial of degree n-r and gα≠0, then fα=f'α=f''α=...=fr-1α=0 and f rα≠0. x-α is a factor of the plolynomial f (x) if fα=0. For a < b, if f (a) and f (b) are of opposite sign, then f (x) = 0 has odd number of roots in the interval (a, b). A polynomial equation f(x) = 0 has exactly n roots equal to α if fα=f'α=f''α=...=fn-1α=0 and f nα≠0. Reality checkQuestion:1
If one root of is 3 + 4i, then the real values of b and c areA) B) C) D)
None of these
As one root is 3 + 4i, the other is 3 − 4i. Hence, the quadratic equation should be as follows:
This equation should be identical with .
Hence, the correct option is A.Question:2
The number of real roots of the equation x2-5x+6sinx=0 in the interval -3π,3π is
Given: x2-5x+6sinx=0 x2-5x+6=0 and sinx=0
Now, sinx = 0 In the interval, sinx = 0 has x=-2π,-π, 0, π, 2π as its solution.
Thus, the given equation has 9 roots. Hence, the correct option is (D).Question:3
The range of λ for which the equation (λ + 10) x2 + (2λ − 20) x + (λ − 5) = 0 has real roots of opposite sign is
The given quadratic equation is (λ + 10) x2 + (2λ − 20) x + (λ − 5) = 0.
The roots are real and of opposite sign.
∴ D ≥ 0 and product of the roots < 0.
Product of roots < 0
λ∈ (− 10, 5) The correcoption is (A).Question:4
If x is real, then the minimum value of is
A) B) C)
Since x is real, discriminant ≥ 0.
Thus, the minimum value of is . Hence, the correct answer is option A.Question:5
If 5α2 = 3α + 1 and 5β2 = 3β + 1 (α ≠ β) , find the equation whose roots are.A)
5x2 − 16x+ 12 = 0B)
12x2 − 16x + 5 = 0C)
5x2 + 12x − 16 = 0D)
12x2 + 5x − 16 = 0
Since 5α2 = 3α + 1 and 5β2 = 3β + 1, (α,β ) are the roots of the quadratic equation 5x2 = 3x + 1, i.e. 5x2 − 3x − 1 = 0.
Sum of the roots =
Product of the roots
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