**Polynomials** with degree 2 are called quadratic polynomials. When this polynomial is equated to zero, we get a quadratic equation.

Its **general form** is **a ^{2 }+ bx + c = 0**, where

**a**,

**b**and

**c**are real numbers and

**a ≠ 0**.

In many real life situations, we deal with quadratic equations.

Suppose, we have to make a table of 50m^{2} area with its length twice as its breadth then,

Let **x** be the breadth of the table Therefore, its length will be **2x**

Since, length × breadth = area

Therefore, x.2x = 50

So, **2x ^{2} = 50 (quadratic equation)**

x^{2 }=25 that gives, x = 5

**Thus the length of that table will be = 2x = 10m and breadth will be 5m.**

**Represent the following statements mathematically:**

**Q.1 Rahul and Nikhil together have 50 chocolates. Both of them lost 5 chocolates each and the product of number of chocolates they have now is 300.Find how many chocolates they actually had?**

**Sol**.

Let, **x** be the number of chocolates Rahul had Then, the total number of chocolates Nikhil had = **(50 – x)** chocolates.

After loosing 5 chocolates Rahul had **(x – 5)** chocolates and Nikhil had **(50 – x – 5)** chocolates.

Now, according to the given condition:

(x – 5) (45 – x) = 300

45x – x^{2 }– 225 + 5x=300

**x ^{2 }– 50x + 525 = 0**

**This is the required quadratic equation.**

** **

**Q2.** **Check whether (x – 7) ^{2} +4 = 2x – 9 is a quadratic equation.**

**Sol.**

On, simplifying the above equation we get:

x^{2 }+ 49 – 14x + 4 = 2x – 9

**x ^{2 }– 16x + 62=0**

General form of quadratic equation is **ax ^{2 }+ bx + c = 0**

Therefore, the given equation is a quadratic equation.

**Q.3 Check whether x(x-7) + 3 = (x + 6) (x – 6) is a quadratic equation.**

**Sol.**

On, simplifying the above equation we get:

x^{2 }– 7x + 3 = x^{2 }– 36

** ****7x – 33 = 0**

**This is not a quadratic equation** because general form of a quadratic equation is ax^{2}+bx+c=0 with a ≠ 0 and in this equation a = 0.

**Q.4** Check whether **(x – 2) ^{3 }= 0** is a quadratic equation.

**Sol.**

On, simplifying the above equation we get:

x^{3 }– 8 – 6x^{2}+12x = 0.

**Degree** of this polynomial equation is **3**,

**Thus, it is ****a cubic polynomial equation not a quadratic equation****.**

** **

**Q.5** **Rahul’s father ****is 27** **years older than him. 3 years from now the product of their ages will be 1224. We would like to find Rohan’s present a****ge.**

**Sol.**

Let the present age of Rahul be **x** **years**

Then, present age of Rahul’s father will be = **(x + 27) years**

Now, their ages after 3 years:

Rahul’s age = **(x + 3) years**

Rahul’s father age = x + 27 + 3 = **(x + 30) years**

According to given condition:

(x+3) (x+30) = 1224

x^{2} + 30x +3x +90 –1224 = 0

** x ^{2 }+ 33x – 1134 = 0**. (Where x is the present age of Rahul in years)

**This is the required quadratic equation. **

** **

**Q.6** **There is a rectangular park of area 860m ^{2}. The length (meters) of the park is three more than twice its breadth (meters). Form a quadratic equation to solve this problem.**

**Sol.**

Let, **x** be **length** (meters) and **y** be **breadth** (meter) of the rectangular plot

Now, according to given condition:-

**x = 3 + 2y ……… (1)**

And

**x.y = 860m ^{2}…… (2)**

Now, substituting equation (1) in (2) we get

(3 + 2y) y = 860

2y^{2 }+ 3y = 860

**2y ^{2 }+ 3y – 860 = 0** where y = breadth (in meters)

**This is the required quadratic equation.**

** **

**Q.7** **A train is covering a distance of 540 km from one city to another city at a uniform speed. If the speed of train had been reduced by 6 km/h, then to cover the same distance the train would have taken 1 hour more. Form a quadratic equation of this situation.**

**Sol.**

Let the train travels at the uniform speed of **x km/hr.**

Therefore,

Time taken to cover 540km =

Now,

When speed is reduced by 6km/h then time taken to cover the same distance =

Now, according to given condition:

540x – 540 ( x – 6 ) = x ( x – 6 )

**x ^{2 }– 6x – 3240 = 0.** Where x= speed of train in km/h

**This is the required quadratic equation.**

** ****Finding roots of quadratic equation by factorization**