Class 10 Mathematics Solutions English Medium

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]