Question

hey friend goodevening☺☺

i need ans of this que
u can solve by quadratic equation

1. the product of two consecutive integer is 240 find the integer

2.a two digit number is such that the product of digits is 12 when 36 is added to the number the digits interchange their places. find the number

3.the sum of the areas of two squares is 468 square metre is the different of their perimeter is 24 find the sides of square

4. the height of right angle 7cm less than its breadth is the hypotenuse is 13 cm find the base and height

5.A train travels 360 km at uniform speed if the speed had been 5 kilometre per hour more it would have taken 1 hour less for the same journey find the original speed of the train

u can solve any que but i need an perfect ans

Answer 1

1-. Let the two consecutive integers be: {a}and {a+1},then:
{a}•{a+1} Simplify.
a2+a=240 Subtract 240 from both sides.
a2+a=240=0 Solve by factoring.
{a-15}•{a+16} Apply the zero product rule.
a-15=0or a+16=0 so that:
a=15 or a= -16
There are two answers to this problem:
1) Consecutive integers: 15 and 16.
2) Consecutive integers: -16 and -15.

2-.


Answer:

Number is 26

Let the unit's digit in the number be x, then as product of digits is 12, the ten's digit in the number is 12x.

Hence, the value of the number is 10×12x+x=120x+x

On reversing the digits, unit's digit becomes ten's digit and ten's digit becomes unit's digit and its value will become

10x+12x

It is apparent that 10x+12x is greater than 120x+x by 36

Hence 10x+12x=120x+x+36

or 9x=120−12x+36

or 9x=108x+36 and dividing each term by 9we get

x=12x+4 and now multiply each by x to get

x2=12+4x or x2−4x−12=0

i.e. x2−6x+2x−12=0

or x(x−6)+2(x−6)=0

i.e. (x+2)(x−6)=0

Hence, x=−2 or x=6

But we cannot have negative number in units place

Hence, 6 is in unit's place and in ten's place we have 126=2

and number is 26

3- Let us say that the sides of the two squares are 'a' and 'b'

Sum of their areas = a^2 + b^2 = 468

Difference of their perimeters = 4a - 4b = 24

=> a - b = 6

=> a = b + 6

So, we get the equation

(b + 6)^2 + b^2 = 468

=> 2b^2 + 12b + 36 = 468

=> b^2 + 6b - 216 = 0

=> b = 12

=> a = 18

4- The sides of the two squares are 12 and 18.
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides. Let ABC be right angled triangle, with altitude = AB, Base = BC & Hypotenuse = AC Given Hypotenuse = AC = 13 cm and Altitude is 7 cm less than base Let base = BC = x cm Altitude = AB = Base – 7 = x – 7 Since ABC is a right angled triangle Using Pythagoras theorem Hypotenuse2 = Height2 + Base2 (AC) 2 = AB2 + BC2 (13)2
= (x – 7)2 + x2 13 × 13
= x2 + 72 – 2(x)(7) + x2 169
= x2 + 49 – 14x + x2 169
= 2x2 – 14x + 49 0
= 2x2 – 14x + 49 – 169 0
= 2x2 – 14x – 120 2x2 – 14x – 120
= 0 2 (x2 – 7x – 60)
= 0 x2 – 7x – 60
= 0/2 x2 – 7x – 60 = 0
We factorize by splitting the middle term method x2 + 5x – 12x – 60 = 0 x (x + 5) – 12 (x + 5) = 0 (x – 12) (x + 5) = 0 So, x = 12 , x = –5 But x cannot be negative as length is not negative ∴ Base = x = 12 cm Altitude = x – 7 = 12 – 7 = 5 cm

5- A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train. Let the speed of train be x km/hr From (1) (x + 5) (360/ " 1" ) = 360 (x + 5) ((360 )/ )
= 360 (x + 5) (360 x )
= 360x x(360 x) + 5(360 x )
= 360 x 360x x2 +5(360) 5x
= 360 x 360x x2 + 1800 5x
= 360 x 360x x2 + 1800 5x 360 x
= 0 x2 5x 360 x + 360x + 1800
= 0 x2 5x + 1800 = 0 0
= x2 + 5x 1800 x2 + 5x 1800 = 0
We factorize by splitting the middle term method x2 + 45x 40x 1800 = 0 x (x + 45) 40 (x + 45) = 0 (x + 45) (x 40) = 0 Hence x = 45, x = 40 are the roots of the equation We know that Speed of train = x So, x cannot be negative x = 40 is the solution So, Speed of train = x = 40 km/hr