QUADRATIC EQUATION
If P(x) is a Quadratic polynomial written in descending order of their degrees , then P(x) = 0 is called a quadratic equation.
The general form of a quadratic equation is
ax²+bx+c = 0
where a, b and c are real numbers and a ≠ 0.
ROOTS OF A QUADRATIC EQUATION
A real number ɑ is said to be a root of the quadratic equation ax²+bx+c = 0 and a ≠ 0 , if a aɑ²+bɑ+c = 0.
We can also say that x = ɑ is a solution of the quadratic equation.e.g. The quadratic equation is
x² - 6x + 8 = 0
If x =2 then 2² - 6(2) + 8 = 0
so x =2 is root of quadratic equation.
SOLUTIONS OF QUADRATIC EQUATIONS
1. BY FACTORISATION
In this method, the middle term of the quadratic equation is spliced after further simplification the given quadratic equation converts into a product of two linear factors then the roots of the equation can be obtained by equating each factor equal to zero.
Let the quadratic equation be ax²+bx+c = 0 and its linear factors are (px + q) and (rx + s) then
ax²+bx+c = (px + q) (rx + s)
Now ax²+bx+c = 0
(px + q)(rx + s) =0
px + q = 0 or rx + s = 0
x = -q/ p or x = -s / r
Thus -q / p and -s / r are the roots of the ax² + bx + c =0 .
2. BY USING THE QUADRATIC FORMULA
Let the general quadratic equation be ax² + bx + c =0 and a ≠ 0. Then solution of this equation can be find using the formula .
x = -b ± √(b² -4ac)
2a
* If ɑ and 𝛃 be considered as roots of the quadratic equation, then
ɑ= -b + √(b² - 4ac) 𝛃 = -b - √(b² - 4ac)
2a 2a
* The quality b² -4ac is called the discriminant of the quadratic equation ax²+bx+c = 0 and is denoted by D.
so D = b² -4ac
NATURE OF ROOTS OF EQUATION
Let D = b² -4ac be the discriminant of the given equation , ax² + bx + c =0 and a ≠ 0.
(i) If D> 0 then the two roots are real and unequal
(ii) If D = 0 then the roots are real and equal
(iii) If D
(iv) The roots are real when D ≥ 0.
* If D> 0 and D is perfect square then roots are rational
* IF D> 0 and D is not a perfect square then roots are irrational
* If one of the root of the quadratic equation is a + √b then its other root is a - √b.
SIGN OF QUADRATIC EQUATION
The sign of the quadratic equation ax²+bx+c is always positive if
(i) a is positive and b² -4ac ≤ 0
(ii) a is negative and b² -4ac ≤ 0
SOME IMPORTANT CONDITIONS FOR THE ROOTS OF QUADRATIC EQUATION
1. Condition for having both the roots positive
(-b / a ) > 0 and (c / a) > 0
2. Condition for having both the roots negative
-b / a
3. Condition for having both roots equal in magnitude but opposite in sign -b / a = 0 or b = 0
4. Condition for having both the roots common of two quadratic equations
ax² + bx + c =0
and a1x² + b1x + c1 = 0
a = b = c
a1 b1 c1
5. Condition for having only one root common of two quadratic equations ax² + bx + c =0 and
a1x² + b1x + c1 =0
x² = -x = 1
b1c2-c1b2 a2c1-c2a1 a1b2-a2b1
6. If the roots of the equation ax² + bx + c =0 are ɑ and 𝜷 then the equation with roots 1/ɑ and 1 / 𝛃 will be cx² + bx + a =0 i.e. for this condition a = c.
7. If the roots of the equation ax² + bx + c =0 are reciprocals of each other i.e. roots
are ɑ and 1 /ɑ then a > c.
SUM AND PRODUCTS OF THE ROOTS
Let ɑ, 𝜷 be the roots of the equation ax² + bx + c =0 and a ≠ 0, then
Sum of the roots = ɑ + 𝜷 = -b / a = - Coefficient of x / Coefficient of x²
Product of the roots =ɑ * 𝜷 = c / a = Constant term / Coefficient of x²
FORMATION OF A QUADRATIC EQUATION
If ɑ + 𝜷 one the roots of quadratic equation, then the quadratic equation will be
x² - (ɑ + 𝜷)x + ɑ*𝜷 = 0
or,
x² -( sum of roots )x + product of roots = 0
SOLUTION OF WORD PROBLEMS INVOLVING QUADRATIC EQUATION
In this section, we will discuss some problems based on practical applications of quadratic equation.
In this type of problems
(i) We first formulae a quadratic equation in variable x with the help of the given condition, whose solution is a solution of the given problem.
(ii) Sometimes it may happen that out of the roots of the quadratic equation only one has a meaning for the problem .
(iii) Some times any root of the quadratic equation, which does not satisfy the condition of the problem will be rejected .
EQUATIONS REDUCIBLE TO QUADRATIC EQUATIONS
The equations which at the out set are not quartic equations but can be reduced to quadratic equations buy suitable substitutions are called equations reducible to quadratic equations.
TYPE 1
ax²ⁿ + bxⁿ +c = 0
Here put xⁿ = y => x²ⁿ = y² and equation reduces to ay² + by + c = 0
TYPE 2
Px + Q / x = R
Here reduce the given equation to a quadratic equation by multiplying both sides by 'x' . So Px² + Q = Rx
TYPE 3
√(x + b) = c or √(a - x²) = bx + c
* Squaring both sides gives a quadratic equation without the radical .
* Solve the factorisation or by quadratic formula.
TYPE 4
Equation of the form
√(ax + b) ± √(cx + d) = c
or √(ax + b) ± √(cx + d) ± √(ex + f) = 0
* Squaring both sides once so that only one term containing radical is obtained.
* Keep only term containing radical on one side and all other terms on the other side .
* Squaring again and solve the quadratic equation so obtained.
TYPE 5
Equation of the type
(x + a)(x + b)(x + c)(x + d) + k = 0
Here k may or may not be zero
* sum of any two constant i.e. a,b,c,d is equal to the sum of the other .
* Multiply the products which satisfy first condition.
* Put first two term containing x², then x and solve it as follows.
TYPE 6
Aˣ⁺ª + A⁻ˣ⁺ˢ = C
Here A = 2, 3, 4 ,....
TYPE 7
Equation of the form
(i) a(x² + 1/x²) + b(x + 1/x) + c = 0
(ii) a(x² + 1/x²) + b (x - 1/x) + c = 0
Put (x + 1/x) = y in case (i) and then find x² + 1/x² similarly , (x - 1/x) = y in case (ii) and then find x² + 1/x² and evaluate as follows.
TYPE 8
Equation of the type
ax⁴+bx³+cx²+bx+a=0
* Coefficient of x^4 and constant term should be same
* Coefficient of x^3 and coefficient of x should be same
* Divide the equation by x^2
* Collect the like term
* Now follow steps of type 7
SYMMETRIC FUNCTIONS OF THE ROOTS
Let ɑ and 𝜷 be the roots of a quadratic equation. An expression in ɑ and 𝜷 which remains same when ɑ and 𝜷 are interchanged is known as a symmetric roots of a quadratic equation in terms of its coefficients we always express it in terms of ɑ - 𝜷 and ɑ𝜷 .
The following results may be useful
INEQUATION
Two real numbers or two algebraic expressions related by the symbols >, , ≤ and ≥ are know as inequalities.
5 2 etc
LINEAR INEQUALITIES
An in equation is said to be linear if each variable occurs in first degree only and there is no term involving the product of the variable.
e.g. ax + b ≤ 0
LINEAR INEQUALITY IN ONE VARIABLE
A linear inequality which has only one variable is called linear inequality in one variable.
e.g. ax + b
LINEAR INEQUALITY IN TWO VARIABLE
A linear inequality which have only two variable is called linear inequality in two variable
e.g. 3x + 11 y
GENERAL RULES TO SOLVE LINEAR INEQUATIONS
* Equal numbers may be added or subtracted from both sides of an inequality with out changing its sign e.g. If a > b , the for any number c.
a + c > b + c
* If both sides of an inequality are multiplied or divided by same positive number then the sign of inequality remains the same, e.g. let a,b and c are any real numbers.
If a>b and a>c then a / c > b / c and ac > bc .
If both sides of an inequality are multiplied or divide by negative number, then the sign of inequality reversed.e.g. Let a, b and c be any real numbers.
If a > b and c
*The square root on both the sides of an in equation cannot be taken in the new take the square root on both sides of an equation.
e.g. x² = 16 => x = ⨦ 4
GRAPHICAL SOLUTION OF LINEAR INEQUATIONS IN TWO VARIABLES
Let in equation be ax + by + c ≥ 0 ,
STEP 1
Consider it as ax + by + c = 0 . Draw the graph of this equation.
STEP 2
Choose any point [if possible (0,0)]
If it satisfied the given in equation then the shaded part of the plane contains the point, otherwise shaded the other part as solution .
QUADRATIC INEQUATIONS
An equation of the form
ax² + bx + c ≥ 0
or ax² + bx + c > 0
or ax² + bx + c ≤ 0
or ax² + bx + c
where a ≠ 0 is called a quadratic in equation in one variable x
* Can be solved graphically or algebraically.
SOLUTION OF QUADRATIC INEQUATION
* Factorise the quadratic in equation .
* If discriminant of b² - 4ac of the corresponding equation ax² + bx + c = 0 then ax² + bx + c will always have distinct linear factor.
* When the product of the two factors is positive or >0 then either both the factor are positive or both are negative.
* When the product of the two factors is negative, then two factors will be of opposite signs.
* If b² - 4ac= 0 then ax² + bx + c will be a perfect square.
* If b² - 4ac
EXERCISES
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