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# introduction-Natures of root Class 10 maths 4. Quadratic Equations - CBSE Notes - Toppers Study

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## introduction-Natures of root Class 10 maths 4. Quadratic Equations - CBSE Notes - Toppers Study

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### introduction-Natures of root class 10 Mathematics Chapter 4. Quadratic Equations

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- Introduction-Natures Of Root Class 10 Maths 4. Quadratic Equations - CBSE Notes - Toppers Study
- Class 10 Ncert Solutions
- Solution Chapter 4. Quadratic Equations Class 10
- Solutions Class 10
- Chapter 4. Quadratic Equations Introduction-Natures Of Root Class 10

## 4. Quadratic Equations

### | introduction-Natures of root |

## introduction-Natures of root Class 10 maths 4. Quadratic Equations - CBSE Notes - Toppers Study

**Quadratic Equations**

**Introduction:**

The equation ax^{2} + bx + c = 0, is the standard form of a quadratic equation, where a, b and c are real numbers and a ≠ 0.

**Example:**

1. 3x^{2 }- 5x = 0,

This equation can be expressed in the form of ax^{2} + 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. 5x^{2} + 2x -7=0,

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

3. 3x^{2} ,

This is single term polynomial i.e mononomial. It can be also expressed in the form of ax^{2} + 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 ax^{2} + bx + c = 0. As the ax^{2} 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 ax
^{2}+ 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. x^{3} + 3x^{2} + 4x + 5, 2x^{3} + 4x, 4x^{3} - 5x^{2} + 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:**

**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 ax
^{2}+ bx + c = 0, a ≠ 0. If ax^{2}+ bx + c = 0, the zeroes of quadratic polynomial ax^{2 }+ bx + c and the roots of the quadratic equation ax^{2}+ bx + c = 0 are the same. - The roots of a quadratic equation ax
^{2}+ bx + c = 0, a ≠ 0 gives;

Where b^{2} - 4ac ≥ 0.

Since b^{2} – 4ac determines whether the quadratic equation ax^{2} + bx + c = 0 has real roots or not, b^{2} – 4ac is called the discriminant of this **quadratic equation **and Discriminant is denoded by capital Letter **D**.

Hence,

D = b^{2} – 4ac,

**Nature of Roots of Quadratic Equations:**

**Nature of Roots:**

**Using Quadratic formula we have **

See here **b ^{2} - 4ac** given in under root.

This valuue **b ^{2}**

**- 4ac**is called Discriminant.

Which is denoted by "D".

∴ D = **b ^{2}**

**- 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 b^{2} - 4ac = 0 or (D = 0)]

Example:

**Solution: **

x^{2} - 6x + 9 = 0

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

Checking for existance of roots,

D = *b ^{2} - 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;

7x^{2} + 2x - 3 = 0

**Solution: **

7x^{2} + 2x - 3 = 0

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

Checking for existance of roots,

D = *b ^{2} - 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

8x^{2} + 5x + 3 = 0

**Solution: **

8x^{2} + 5x + 3 = 0

*a = 8, b = 5, c = 3*

Checking for existance of roots,

D = *b ^{2} - 4ac*

D = *(5) ^{2} - 4 × 8 × 3*

D = *25 - 96*

D = *-71*

Hence D < 0

∴ There is no roots [Nature-III]

##### Other Pages of this Chapter: 4. Quadratic Equations

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