**Quadratic Equations***Standard form of equation is*

*ax*^{2}+ bx + c = 0*where a, b and c are the co-efficients and a ≠ 0*

**Q1: Check whether the following are quadratic equations :**

**(i) (x + 1)**

^{2}= 2(x – 3)**(ii) x**

^{2}– 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) x**

^{2}+ 3x + 1 = (x – 2)^{2}Answer:

(i) (x + 1)

^{2}= 2(x – 3)

⇒ x

^{2}+ 2x + 1 = 3x - 6

⇒ x

^{2}+ (2-3)x + 1 + 6 = 0

⇒ x

^{2}- x + 7 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0(ii) x

^{2}– 2x = (–2) (3 – x)

⇒ x

^{2}– 2x = (–2 × 3) + ( –2 × –x)

⇒x

^{2}– 2x = –6 + 2x

⇒ x

^{2}– 2x - 2x + 6 = 0

⇒ x

^{2}– 4x + 6 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

⇒ x

^{2}+ x -2x - 2 = x

^{2}+ 3x -x -3

⇒ x

^{2}- x - 2 = x

^{2}+ 2x - 3

⇒ x

^{2}- x - 2 - x

^{2}- 2x + 3 = 0

⇒ -3x + 1 = 0

⇒ 3x - 1 = 0

Since the equation is NOT of the form

*ax*, it is NOT a quadratic equation.

^{2}+ bx + c = 0(iv) (x – 3)(2x +1) = x(x + 5)

⇒ 2x

^{2}+ x - 6x - 3 = x

^{2}+ 5x

⇒ 2x

^{2}- 5x - 3 - x

^{2}- 5x = 0

⇒ x

^{2}- 10x - 3 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0(v) (2x – 1)(x – 3) = (x + 5)(x – 1)

⇒ 2x

^{2}- 6x - 1x + 3 = x

^{2}- 1x + 5x - 5

⇒ 2x

^{2}- 7x + 3 = x

^{2}+ 4x - 5

⇒ 2x

^{2}- 7x + 3 - x

^{2}- 4x + 5 = 0

⇒ x

^{2}- 11x + 8 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0(vi) x

^{2}+ 3x + 1 = (x – 2)

^{2}

⇒ x

^{2}+ 3x + 1 = x

^{2}+ 4 - 4x

⇒ x

^{2}+ 3x + 1 - x

^{2}- 4 + 4x =0

⇒ 7x - 3 = 0

Since the equation is NOT of the form

*ax*, it is NOT a quadratic equation.

^{2}+ bx + c = 0**Q2: Represent the following situations in the form of quadratic equations:**

**(i) The area of a rectangular plot is 528 m**

^{2}. 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.Answer: Let the breadth of the rectangular plot = b

*m*

∴ Length of the plot = (2 × b + 1)

*m =*2b + 1

Area = length × breadth = 528

*m*

^{2}⇒ (2b+1) × b = 528

⇒ 2b

^{2}+ b =528⇒ 2b

^{2}+ b - 528 = 0

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

Answer: Let the first integer number be = p

The next consecutive positive integer = p+1

Product = p × (p +1) = 306

⇒ p

^{2}+ p = 306

⇒ p

^{2}+ p - 206 = 0

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

Answer: Let Rohan's age = t years

Rohan's mother age = t + 26 years

3 years from now, Rohan's age = t + 3

Age of Rohan's mother after 3 years = t + 26 + 3 = t + 29

Product of their ages after three years = (t + 3)(t + 29) = 360

⇒ t

^{2}+ 29t + 3t + 87 = 360

⇒ t

^{2}+ 32t + 87 - 360 = 0

⇒ t

^{2}+ 32t - 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 the speed of the train = v

*km/h*

∵ Distance = speed × time

Time taken by train (t) = 480/v

*h*

If speed decreases by 8 km/h, new speed = v - 8

*km/h*

Time taken to cover 480 km with new speed = t + 3 hours = 480/v + 3 hours

∵ Distance = speed × time

⇒ (v - 8) × (480/v + 3) = 480

⇒ 480 + 3v - 3840/v - 24 = 480

⇒ 480 + 3v - 3840/v - 24 - 480 = 0

⇒ 3v - 3840/v - 24 = 0

⇒ 3v

^{2}- 3840 - 24v = 0

⇒ v

^{2}- 8v - 1280 = 0

**Q3(FA Manual): Which of the following are quadratic equations?**

**(i) √x - x = 4**

**(ii) x + 1/x = 5**

**(iii) 7 v**

^{2 }= 49Answer:

(i) √x - x = 4

⇒ x

^{1/2}- x = 4

Since Since the equation is NOT of the form

*ax*, it is NOT a quadratic equation.

^{2}+ bx + c = 0(ii) x + 1/x = 5

⇒ x + 1/x - 5 = 0

Multiply by x to both sides,

⇒ x

^{2}+ 1 - 5x = 0

⇒ x

^{2}-5x + 1 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0(iii) 7 v

^{2 }= 49

⇒ 7v

^{2 }- 49 = 0

⇒ 7v

^{2}+ 0v - 49 = 0

Since the equation is of the form

*ax*, it is a quadratic equation.

^{2}+ bx + c = 0[Note: Equation is quadratic, when the first term must be

*ax*and a ≠ 0 but coefficients b and c can be zero].

^{2}.

Thanks for your grateful informations, this blogs will be really help for chemistry notes.

ReplyDeletein 1st question there is some mistake. instead of 3x-6 it should be 2x-6.

ReplyDeleteYes,it should be 2x-6.

Delete