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## Quadratic Polynomials

A polynomial of the form of p(x) = ax^{2} + bx + c is known as quadratic polynomial, where a, b, c are real and a ≠ 0.

For example: p(x) = 3x^{2} + 5x – 2, here a = 3, b = 5 and c = -2.

## Quadratic Equations

An equation in the form of ax^{2} + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

For example: 3x^{2} + 5x – 2 = 0, here a = 3, b = 5 and c = -2.

## Solving Quadratic Equations by Factorization

To solve a quadratic equation by factorization, we do the following steps:

Step 1 : Express the equation in the form of ax^{2} + bx + c = 0.

Step 2 : Factorize ax^{2} + bx + c using middle term splitting method.

Step 3 : Put each factor equal to zero.

Step 4 : Solve each equation.

## Equations Reducible to Quadratic Equations

An equation in the form of ax^{2} + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

In some questions, the given equations are not in form of ax^{2} + bx + c = 0. Then, to solve such type of equations, first we need to reduce the given equation in form of quadratic equation ax^{2} + bx + c = 0 by using suitable algebraic transformation and then solve it.

## Solving Quadratic Equation by Quadratic Formula

An equation in the form of ax^{2} + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

To solve quadratic equations, there are two methods :

2. Using Quadratic Formula

In this exercise, the quadratic equations are solved by using the quadratic formula.

## Proof of Quadratic Formula

Let the quadratic equation is ax^{2} + bx + c = 0, where a ≠ 0. We need to find the value of x.

ax^{2} + bx + c = 0

⇒ ax^{2} + bx = – c [Transposing constant term c to RHS]

⇒ x^{2} + (b/a)x = – c/a [Dividing both sides by a to make coefficient of x^{2} = 1]

⇒ x^{2} + (b/a)x + b^{2}/4a^{2 }= b^{2}/4a^{2 }– c/a [Adding both sides by b^{2}/4a^{2}]

⇒ x^{2} + 2(b/2a)x + b^{2}/4a^{2 }= (b^{2 }– 4ac)/4a^{2}

⇒ (x + b/2a)^{2 }= (b^{2 }– 4ac)/4a^{2}

⇒ (x + b/2a)^{ }= ± √(b^{2 }– 4ac)/2a

⇒ x = – b/2a ± √(b^{2 }– 4ac)/2a

⇒ x = – b ± √(b^{2 }– 4ac)]/2a.

Therefore, if the quadratic equation ax^{2} + bx + c = 0, then the value of x is x = [- b ± √(b^{2 }– 4ac)]/2a. This is known as Quadratic Formula.

Hence, there are two values of x as x = [- b + √(b^{2 }– 4ac)]/2a or x = [- b + √(b^{2 }– 4ac)]/2a.

## Discriminant of Quadratic Equations

^{2} + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

Then the value of x is x = [- b ± √(b^{2 }– 4ac)]/2a. This is known as Quadratic Formula.

Hence, there are two values of x as x = [- b + √(b^{2 }– 4ac)]/2a or x = [- b + √(b^{2 }– 4ac)]/2a.

Now, in x = [- b ± √(b^{2 }– 4ac)]/2a, the term under square root, that is (b^{2 }– 4ac) is known as the discriminant of the quadratic equation of ax^{2} + bx + c = 0 and it is denoted by D.

Thus, **D = b ^{2 }– 4ac.**

## Nature of Roots of Quadratic Equations

The nature of the roots determines the types of the roots of quadratic equation of ax^{2} + bx + c = 0, whether the roots are real or imaginary and if the roots are real, then it will be rational or irrational and equal or unequal.

Since D = b^{2 }– 4ac is in quadratic formula x = [- b ± √(b^{2 }– 4ac)]/2a, then the value of the discriminant D = b^{2 }– 4ac determines the nature of the roots of quadratic equations.

You don’t need to solve the quadratic equation of ax^{2} + bx + c = 0. You just need to find the value of its discriminant D = b^{2 }– 4ac.

## How To Determine Nature of Roots of Quadratic Equations

If the discriminant D = b^{2 }– 4ac is **positive** (means, D > 0), then the roots of quadratic equation are **REAL **and** DISTINCT**.

If the discriminant D = b^{2 }– 4ac is **positive** and **perfect square**, then the roots of quadratic equation are **REAL**, **RATIONAL** and **DISTINCT**.

If the discriminant D = b^{2 }– 4ac is **positive** and **NOT a perfect square**, then the roots of quadratic equation are **REAL**,** IRRATIONAL** and **DISTINCT**.

If the discriminant D = b^{2 }– 4ac is **zero** (means, D = 0), then the roots of quadratic equation are **REAL** and **EQUAL**. And, Root will be α = β = -b/2a.

If the discriminant D = b^{2 }– 4ac is **negative** (means, D < 0), then the roots of quadratic equation are **NOT REAL** , it means the roots are **IMAGINARY**.

Summary of the nature of the roots of quadratic equation is as below.

### Quadratic Equations Word Problems

In word problems, first we need to make a quadratic equation according to the given question and then we need to solve them.

S Chand ICSE Class 10 Maths Chapterwise Notes & Solutions |
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Chapter 1 : GST Notes & Solutions | Notes | Exercise 1 |

Chapter 2 : Banking Notes & Solutions| Notes | Exercise 2 | Revision Exercise |

Chapter 3 : Shares & Dividends Notes & Solutions| Notes | Exercise 3A | Exercise 3B | Revision Exercise |

Chapter 4 : Linear Inequations in One Variable Notes & Solutions| Notes | Exercise 4 | Revision Exercise |

Chapter 5 : Quadratic Equations Notes & Solutions| Notes | Exercise 5A | Exercise 5B | Exercise 5C| Exercise 5D| Exercise 5E| Revision Exercise |

Chapter 6 : Ratios & Proportions Notes & Solutions| Notes | Exercise 6A | Exercise 6B | Exercise 6C| Revision Exercise |