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Quadratic Equations

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Introduction:

The equation ax² + bx + c = 0, is the standard form of a quadratic equation, where a, b and c are real numbers and a ≠ 0. 

Example:

1. 3x² - 5x = 0, 

This equation can be expressed in the form of ax² + bx + c = 0. then 

a = 3, b = -5, c = 0, 

Here c = 0, As Term c is disappear. 

This also showing a ≠  0.  Hence this is a quadratic equation. 

2. 5x² + 2x -7=0, 

Here a = 5, b = 2, c = -7, so it can be also expressed in the form of ax² + bx + c =0, 

3. 3x² ,

This is single term polynomial i.e mononomial. It can be also expressed in the form of ax² + bx + c = 0. In which a= 3, b = 0, c = 0,  Here b = 0, c = 0 but there is no a ≠  0. 

So, this is also a quadratic equation. 

4. 4x + 9, 

This cannot be expressed in the form of ax² + bx + c = 0. As the ax² term is disappear. Hence a = 0. Which can not fulfill the condition of to be a quadratic equation. 

• All quadratic polinomials can be expressed in the form of quadratic equation ax² + bx + c = 0. 
• ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.

Another equations which are not a quadratic equation.

1. x³ + 3x² + 4x + 5, 2x³ + 4x, 4x³ - 5x² + 7 and all cubic polynomials. 

2. All linear equations like 4x + 3, 5x, 7x + 2 etc.  

4. Polynomials of power more than 2 and less than 2. 

Nature of Roots:

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Roots of Quadratic equations:

• Each quadratic equation has two roots. they are said to be α and β. 
• A real number α is said to be a root of the quadratic equation ax² + bx + c = 0, a ≠ 0. If ax² + bx + c = 0, the zeroes of quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.
• The roots of a quadratic equation ax² + bx + c = 0, a  ≠  0 gives;

         

         Where  b² - 4ac ≥ 0. 

Since b² – 4ac determines whether the quadratic equation ax² + bx + c = 0 has real roots or not, b² – 4ac is called the discriminant of this quadratic equation and Discriminant is denoded by capital Letter D. 

Hence, 

D = b² – 4ac,

Nature of Roots of Quadratic Equations:

Nature of Roots:

Using Quadratic formula we have 

See here b² - 4ac given in under root. 

This valuue b² - 4ac is called Discriminant.

Which is denoted by "D".

∴ D = b² - 4ac

[ Nature of root is determined by the value of Discriminant;]

There are three natures of roots.

(a)  D = 0; [Two equal and real roots, if b² - 4ac = 0 or (D = 0)]

Example:  

Solution: 

x² - 6x + 9 = 0

a = 1, b = -6, c = 9

Checking for existance of roots,

D = b² - 4ac

D = (-6)² - 4 × 1 × 9

D = 36 - 36

D = 0

Hence D = 0

∴ There is two equal and real roots [Nature-I ]

This equation gives two equal and real roots x = 3, and x = 3. 

Such equation which have equal and real root is also called a complete square equation. 

(b) D > 0; [ Two real and distinct root]

Example;

7x² + 2x - 3 = 0

Solution: 

7x² + 2x - 3 = 0

a = 7, b = 2, c = -3

Checking for existance of roots,

D = b² - 4ac

D = (2)² - 4 × 7 × -3

D = 4 - (-84)

D = 4 + 84

D = 88

Hence D > 0

∴ There is two real and distinct roots [Nature-II]

(c) D < 0; No Real roots

Example

8x² + 5x + 3 = 0

Solution: 

 8x² + 5x + 3 = 0

a = 8, b = 5, c = 3

Checking for existance of roots,

D = b² - 4ac

D = (5)² - 4 × 8 × 3

D = 25 - 96

D = -71

Hence D < 0

∴ There is no roots [Nature-III]

Quadratic Equations
Questions for practice

1 Marks questions;

 

Q1.   Write the discriminant of the quadratic equation 3x² - 5x - 11 = 0.

Q2.   For what value of k quadratic equation x² - kx + 4 = 0 has equal

         roots. 

Q3.   Write the nature of the roots of equation 4x² - 12x + 8 = 0. 

Q4.   What will be the value of D when given quadratic equation is a

         complete square. 

Q5.   Find the value of x in equation x² - 3 = 0.

Q6.   Write the value of x in equation x² - 15 = 0.   

Q7.   If x = - 2 is a root of the quadratic equation 3x² - 5x + 2k = 0. Then

         find the value of k. 

Q8.   For what value of k for which x = 3 is a solution of kx² - 2x - 1 = 0. 

Q9.   For what value of p the quadratic equation 2x² - 6x + p = 0 has

         real and distinct roots.  

Q10.  If one root of the equation x² + 7x + k = 0 is - 2, then find the value

          of k and other root. 

Q11.  For what value of k the given quadratic equation will be a

          complete square. 

          

Q12.  Find the solution of quadratic equation ax² – 2abx = 0. 

Q13.  Write a quadratic equation whose two roots are - 3 and 4. 

Q14.  Find the nature of equation (x – 2a ) (x – 2b ) = 4ab. 

Q15.  What is the middle term of a quadratic equation which roots are 

          - 4 and - 5 ?

Q16.  2 is the one of the root of quadratic equation 3x² - 11x + p = 0 then

          find the value of p. 

Q17.  Write the discriminant of the quadratic equation . 

          

Q18.  What is the nature of this equation. 

          

Q19.   For what value of k the equation x² + kx + 4 = 0 has no real roots?

Q20.   For what value of p the equation

           

           has real roots?

Q21.  Find the roots of given quadratic equation by the method of 

           completing the square. 

          

Q22.  Find the value of a nad b such that x = 1, x = - 2 are the solution

          of the quadratic equation x² + ax + b = 0.

Q23.   Solve for x

           abx² + (b² - ac) x - bc = 0

Q24.   solve for x

           3y² + (6 + 4a)y + 8a = 0

Q25.  Divide 51 in to two parts such that their product is 378.

Q26.  if the roots of the equation (b-c)x² + (c - a)x + (a - b) = 0 are equal

          then prove that 2b = a + c. 

Q27.  Find the value of k from the given equation so that equation 

          has equal roots. 

         

Q28.  Product of two number is 36 and difference is 9, find those number.

Q29.  Find the value of k if the equation has two equal and real roots.

 

Q30. Find the nature of following quadratic equations: 

(A)  x² + 4x + 9 = 0

(B)  3x² - 2x - 8 = 0

(C)  11x² + 8x - 3 = 0

(D)  x² + 7x + 4 = 0

(E)  x² + 4x + 4 = 0

(F)  3x² + 10x + 8 = 0

(G) 4x² + 3x + 5 = 0

(H)  7x² + 12x + 5 = 0

(I)  5x² - 10x + 5 = 0

Long-Answered Questions:

Q31. A journey of 192km from Mumbai to Pune takes 2hrs less by a fast train then by a slower train. If the speed of the slower train is 16 km/hr less than the faster train, find the speed of each train.

Q32. A part of monthly expenses of a family is constant and the remaining varies with the price of wheat. When the rate of wheat is Rs.250 per quintal, the total monthly expense if the family is Rs.1000 and when it is Rs.240 per quintal, the total monthly expense is Rs.980. Find the total monthly expenses of the family when cost wheat is Rs.350 per quintal.

Q33. A man travels 600km partly by train and partly by car. If he covers 400km by train and rest by car, it takes him 6hrs and 30minutes. But if he travels 200km by train and rest by car, he takes half an hour more. Find the speed of the train and that of the car.