Solve the followings Questions.

1. Check whether the following are quadratic equation?
(i) (x+1)² = 2(x-3)
(ii) x²-2x = (-2)(3-x)
(iii) (x − 2) (x + 1) = (x − 1) (x + 3)
(iv) (x − 3) (2x + 1) = x (x + 5)
(v) (2x − 1) (x − 3) = (x + 5) (x − 1)
(vi) x²+3x+1 = (x-2)²
(vii) (x+2)³ = 2x(x²-1)
(viii) x³-4x²-x+1 = (x-2)³

Answer:
An equation of the form ax²+bx+c = 0, a ≠ 0 and a, b, c are real numbers is called quadratic equation.

(i) (x+1)² = 2(x-3)
x²+2x+1 = 2x-6
x²+2x+1 -2x+6 = 0
x²+7 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1≠ 0, b = 0, c = 7
Hence the given equation is quadratic equation.

(ii) x²-2x = (-2)(3-x)
x²-2x  = -6+2x
x²-2x -2x+6 = 0
x²-4x+6 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1≠ 0, b = -4, c = 6
Hence the given equation is quadratic equation.

(iii) (x-2)(x+1) = (x-1)(x+3)
x²-2x+x-2 = x²-x+3x-3
x²-2x+x-2 -x²+x-3x+3= 0
-3x+1 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = -3, c = 1
Hence the given equation is not quadratic equation.

(iv) (x-3)(2x+1) = x(x+5)
x²-6x+x-3 = x²+5x
x²-6x+x-3 -x²-5x = 0
-10x-3 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = -10, c = -3
Hence the given equation is not quadratic equation.

(v) (2x-1)(x-3) = (x+5)(x-1)
2x²-6x-x+3 = x²+5x-x-5
2x²-6x-x+3 -x²-5x+x+5 = 0
x²-11x+8 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1 ≠ 0, b = -11, c = 8
Hence the given equation is quadratic equation.

(vi) x²+3x+1 = (x-2)²
x²+3x+1 = x²-4x+4
x²+3x+1 -x²+4x-4= 0
7x-3 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = 7, c = -3
Hence the given equation is not quadratic equation.

(vii) (x+2)³ = 2x(x²-1)
x³+6x²+12x+8 = 2x³-2x
x³+6x²+12x+8 -2x³+2x = 0
x³ – 6x²-14x-8 = 0
It is not in the form of ax²+bx+c = 0s
Hence the given equation is not a quadratic equation.

(viii) x³-4x²-x+1 = (x-2)³
x³-4x²-x+1 = x³-6x²+12x-8
x³-4x²-x+1 -x³+6x²-12x+8= 0
2x²-13x+9 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 2 ≠ 0, b = -13, c = 9
Hence the given equation is quadratic equation.
 

Represent the following situations in the form of quadratic equations:

2. (i) The area of rectangular plot is 528 m². The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot .

Answer:

Let the breadth of the plot is x meters
Then, the length is (2x + 1) meters.
Area = length × breadth
528 = x × (2x + 1)
528 = 2x² + x
2x² + x – 528 = 0

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

Answer:

Let x and x+1 be two consecutive integers.
Then, x(x + 1) = 306
x² + x – 306 = 0

(iii) Rohan’s mother is 26 years older than him. The products of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

Answer:

Let x be the Rohan’s present age. Then, his mother’s present age is x+26
After 3 years theirs ages will be x+3 and x+26+3 = x+29 respectively.
Hence,
(x+3)(x+29) = 360
x²+3x+29x+87 = 360
x²+32x- 273 = 0

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Answer:

Let speed of train be x km/h

Time taken by train to cover 480 km = 480xhours

If, speed had been 8km/h less then time taken would be (480x−8) hours

According to given condition, if speed had been 8km/h less then time taken is 3 hours less.

Therefore, 480x – 8 = 480x + 3

⇒480 (1x – 8 − 1x) = 3

⇒480 (x – x + 8) (x) (x − 8) = 3

⇒480 × 8 = 3 (x) (x − 8)

This is a Quadratic Equation.

CHAPTER NAME	OLD NCERT	NEW NCERT
Real Numbers	EXERCISE 1.1
	EXERCISE 1.2	1.1	CLICK HERE
	EXERCISE 1.3	1.2	CLICK HERE
	EXERCISE 1.4
Polynomials	EXERCISE 2.1	2.1	CLICK HERE
	EXERCISE 2.2	2.2	CLICK HERE
	EXERCISE 2.3
	EXERCISE 2.4
Pair of Linear Equations in Two Variables	EXERCISE 3.1
	EXERCISE 3.2	3.1	CLICK HERE
	EXERCISE 3.3	3.2	CLICK HERE
	EXERCISE 3.4	3.3	CLICK HERE
	EXERCISE 3.5
	EXERCISE 3.6
	EXERCISE 3.7
Quadratic Equations	EXERCISE 4.1	4.1	CLICK HERE
	EXERCISE 4.2	4.2	CLICK HERE
	EXERCISE 4.3
	EXERCISE 4.4	4.3	CLICK HERE
Arithmetic Progressions	EXERCISE 5.1	5.1	CLICK HERE
	EXERCISE 5.2	5.2	CLICK HERE
	EXERCISE 5.3	5.3	CLICK HERE
	EXERCISE 5.4	5.4 (Optional)	CLICK HERE
Triangles	EXERCISE 6.1	6.1	CLICK HERE
	EXERCISE 6.2	6.2	CLICK HERE
	EXERCISE 6.3	6.3	CLICK HERE
	EXERCISE 6.4
	EXERCISE 6.5
	EXERCISE 6.6
Coordinate Geometry	EXERCISE 7.1	7.1	CLICK HERE
	EXERCISE 7.2	7.2	CLICK HERE
	EXERCISE 7.3
	EXERCISE 7.4
Introduction to Trigonometry	EXERCISE 8.1	8.1	CLICK HERE
	EXERCISE 8.2	8.2	CLICK HERE
	EXERCISE 8.3
	EXERCISE 8.4	8.3	CLICK HERE
Some Applications of Trigonometry	EXERCISE 9.1	9.1	CLICK HERE
Circles	EXERCISE 10.1	10.1	CLICK HERE
	EXERCISE 10.2	10.2	CLICK HERE
Construction
Areas Related to Circles	EXERCISE 12.1
	EXERCISE 12.2	11.1	CLICK HERE
	EXERCISE 12.3
Surface Areas and Volumes	EXERCISE 13.1	12.1	CLICK HERE
	EXERCISE 13.2	12.2	CLICK HERE
	EXERCISE 13.3
	EXERCISE 13.4
	EXERCISE 13.5
Statistics	EXERCISE 14.1	13.1	CLICK HERE
	EXERCISE 14.2	13.2	CLICK HERE
	EXERCISE 14.3	13.3	CLICK HERE
	EXERCISE 14.4
Probability	EXERCISE 15.1	14.1	CLICK HERE
	EXERCISE 15.2