EXERCISE 4.1
1.Check whether the following are Quadratic Equations.
(i) (x + 1)2 = 2 (x − 3)
Solution: (x + 1)2 = 2 (x −3) use (a+b)2 = a2 + b2 + 2ab
x² +2 x +1 = 2 x − 6
x² +2 x +1 – (2 x − 6) = 0
x² +2 x +1 , 2 x + 6 = 0
x² + 7 =0 .Here, degree of equation is 2.
Since the above equation is in the form of ax2 + bx + c = 0.
∴ It is a quadratic equation.
(ii) x2 −2x = (−2) (3 − x)
Solution: x2 −2 x = −6 + 2 x
x2 − 2 x + 6 −2 x = 0
x2 − 4x + 6 = 0 Here, degree of equation is 2.
Since the above equation is in the form of ax2 + bx + c = 0.
∴ It is a quadratic equation.
(iii) (x − 2) (x + 1) = (x − 1) (x + 3)
Solution: ( x − 2) (x + 1) = (x − 1) (x + 3)
⇒ x2 – x – 2 = x2 + 2x – 3
⇒ x² −x² −x−2x −2 +3 = 0
⇒ −3x + 1 = 0
Here, degree of equation is 1.
Therefore, it is not a quadratic equation.
(iv) (x − 3) (2x + 1) = x (x + 5)
Solution:: (x − 3) (2x + 1) = x (x + 5)
= 2 x2 + x −6x −3 = x2 + 5x
= 2 x2 + x −6x −3 − x2 − 5x = 0
= x2 −10x − 3 = 0
Here, degree of equation is 2.
Therefore, it is a quadratic equation.
(v) (2x − 1) (x − 3) = (x + 5) (x − 1)
Solution: (2x − 1) (x − 3) = (x + 5) (x − 1)
2 x²−6x − x + 3 = x2 − x + 5x − 5
2 x² −7x + 3 − x2 + x − 5x + 5 = 0
x2 −11x + 8 = 0
Here, degree of equation is 2.
Therefore, it is a quadratic equation.
(vi) x2 + 3x + 1 = (x− 2)
Solution: By using the formula for (a+b)2 = a2 + 2ab + b2
⇒ x2 + 3x + 1 = x2 + 4 – 4x
⇒ x2 + 3x + 1 − x2 −4 + 4x = 0
⇒ 7x – 3 = 0
Since the above equation is not in the form of ax² + bx + c = 0.
Therefore,it is not a quadratic equation.
(vii) (x + 2)³ = 2x (x2 – 1)
Solution: (x + 2)³ = 2x (x2 – 1)
By using the formula for (a+b)³ = a³ + 3a²b +3ab² + b³
x3 + 8 +6x2 + 12x = 2 x3 – 2x
⇒ x3 + 8 +6x2 + 12x − 2 x3 + 2x = 0
⇒ −x3 + 14x + 6x2 + 8 = 0
Since the above equation is not in the form of ax2 + bx + c = 0.
Therefore, it is not a quadratic equation.
(viii) x³ – 4x2 – x + 1 = (x −2)2
Solution:: x – 4x2 – x + 1 = (x −2)2
By using the formula for (a + b)2 = a2 + 2ab + b2
⇒ x3– 4x2 – x + 1 = x3 – 8 – 6x2 + 12x
⇒ x3– 4x2 – x + 1 − x3 + 8 + 6x2 − 12x = 0
⇒ 2x2– 13x + 9 = 0. Here, degree of equation is 2.
Since the above equation is in the form of ax² + bx + c = 0.
Therefore, it is quadratic equation.
2. Represent the following situations in the form of Quadratic Equations:
(i) The area of rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
Solution: We are given that area of a rectangular plot is 528 m2.
Let breadth of rectangular plot be x metres
Length is one more than twice its breadth.
Therefore, length of rectangular plot is (2x + 1) metres
Area of rectangle = length × breadth
⇒ 528 = x (2x + 1)
⇒ 528 = 2x2 + x.
⇒ 2x2 + x – 528 = 0
This is a Quadratic Equation.
(ii) The product of two consecutive numbers is 306. We need to find the integers.
Solution: Let two consecutive numbers be x and (x + 1).
It is given that x (x + 1) = 306
⇒ x2 + x = 306
⇒ x2 + x – 306 = 0
This is a Quadratic Equation.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) after 3 years will be 360. We would like to find Rohan’s present age.
Solution: Let present age of Rohan = x years
Let present age of Rohan’s mother = (x + 26) years
Age of Rohan after 3 years = (x + 3) years
Age of Rohan’s mother after 3 years = x + 26 + 3 = (x + 29) years
According to given condition:
(x + 3) (x + 29) = 360
⇒ x2 + 29x + 3x + 87 = 360
x2 + 32x − 273 = 0
This is a Quadratic Equation
(iv) A train travels a distance of 480 km at uniform speed. If, the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. We need to find speed of the train.
Solution: Let speed of train be x km/h
Time taken by train to cover 480 km = 480x hours
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)
⇒ 3840 = 3x2 + 24x
⇒ 3x2 + 24x −3840 = 0
Dividing the equation by 3, we get
x2 + 8x −1280 = 0
This is a Quadratic Equation.
Additional Questions:.
1.Check whether the following are Quadratic Equations.
(a) x2 + 7x – 12 = 0 (b) 9x2 – 12 x = 0
2. Represent the following situations in the form of Quadratic Equations:
(a)The sum of the ages of a woman and her daughter is 40 years. The product of their ages five years ago was 125 years. Find their present ages.
(b) A plane left 30 minutes later than the schedule time and in order to reach its destination 1500 km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.