第 10 类 RD Sharma 解决方案 - 第 8 章二次方程 - 练习 8.7 |设置 1

问题 1：找出两个平方和为 85 的连续数。

解决方案：

Let first number is = x

⇒ Second number = (x+1)

Now according to given condition—

⇒ Sum of squares of the numbers = 85

⇒ x² + (x+1)² = 85

⇒ x² + x² + 2x + 1 = 85         [ because (a+b)² = a² + 2ab + b²]

⇒ 2x² + 2x + 1 – 85 = 0        

⇒ 2x² + 2x – 84 = 0

⇒ x² + x – 42 = 0                 [ dividing by 2 both sides]

now for factorization, convert coefficient of x in difference form of two numbers such that product of those numbers

be 42-

⇒ x² + (7-6)x – 42 = 0

⇒ x² + 7x -6x -42 = 0

⇒ x(x+7) – 6(x+7) = 0

⇒ (x+7)(x-6) = 0

⇒ either     x+7 = 0              or                x-6 = 0

                   x = -7               or                 x = 6

Now when   x = -7

⇒ First number = x = -7              and       Second number = x+1 = -7+1

                                                                                                      = -6

So numbers are -7, -6.

Now when x = 6

⇒ First number = x = 6                and       second  number = x+1 = 7

So numbers are 6, 7.              

问题 2：将 29 分成两部分，使得两部分的平方和为 425。

解决方案：

Let first part is = x

so second part will be = (29 – x)

Now coming to the condition-

⇒ x² + (29-x)² = 425

⇒ x² + 292 – 2*29*x + x² = 425              [because (a+b)² = a² + 2ab + b²]

⇒ 2x² + 841 – 58x = 425                             [because 29²=841]

⇒ 2x² -58x + 841-425 = 0

⇒ 2x² – 58x + 416 = 0

⇒ x² – 29x + 208 = 0

by factorization method—

⇒ x² – (16+13)x + 208 = 0

⇒ x² -16x – 13x + 208 = 0

⇒ x(x-16) – 13(x-16) = 0

⇒ (x-16)(x-13) = 0

Either      x-16 = 0                     or                 x-13 = 0

               x = 16                        or                    x = 13

when first part = 16  then second part = 29 – x

                                                             = 29-16

                                                             = 13

and when first part = 13 then second part = 29-13

                                                                  = 16

So parts will be 13, 16.

问题 3：两个正方形的边分别为 x cm 和 (x + 4) cm。它们的面积之和为656 cm ² 。找到正方形的边。

解决方案：

It is given that-

the side of first square = x cm

and that of second is = (x+4) cm

And we know that area of a square = (side)²

so area of first square = x²

and  area of second square = (x+4)²

Now according to the given condition—

⇒ (Area of first square) + (Area of second square) = 656

⇒ x² + (x+4)² = 656

⇒ x² + x² + 2*x*4 + 42 = 656               [because (a+b)² = a² + 2*a*b + b²]

⇒ 2x² + 8x + 16 – 656 = 0

⇒ 2x² + 8x – 640 = 0

⇒ x² + 4x – 320 = 0                  [dividing by 2 both sides]

By factorization method—

⇒ x² +(20-16)x – 320 = 0

⇒ x² + 20x -16x – 320 = 0

⇒ x(x+20) – 16(x+20) = 0

⇒ (x+20)(x-16) = 0

Either     x+20=0               or          x-16 = 0

              x = -20               or              x=16

but x = -20 is invalid because length can never be negative,

So on discarding x=-20 and taking x=16   —

side of first square is = x = 16 cm

and the side of second square is = x+4

                                                   = 20 cm

问题 4：两个数之和是 48，它们的乘积是 432。找出这些数。

解决方案：

Let the first number = x

So second number = (48 – x)      [because sum of numbers is 48]

Now it is also given-

Product of number is = 432

⇒ x*(48-x) = 432

⇒ 48x -x² = 432

⇒ x² – 48x + 432 = 0

By factorization method–

⇒ x² – (36+12)x + 432 = 0

⇒ x² – 36x – 12x + 432 = 0

⇒ x(x-36) – 12(x-36) = 0

⇒ (x-36)(x-12) = 0

Either    x-36 = 0          or            x-12 = 0

                x = 36         or             x = 12

When x=36 then –

First number = x = 36

and second number = 48-x = 12

And when x=12 then –

First number = x = 12

and second number = 48-x = 36

⇒ Means One number is 12 and another is 36.

问题 5：如果将一个整数与其平方相加，则和为 90。借助二次方程求整数。

解决方案：

Let the number is = x

So it’s square is = x²

Now according to the given condition-

Number + Square of number = 90

⇒ x + x² = 90

⇒ x² + x – 90 = 0

By factorization method-

⇒ x² +(10-9)x – 90 = 0

⇒ x² + 10x – 9x – 90 = 0

⇒ x(x+10) – 9(x+10) = 0

⇒ (x+10)(x-9) = 0

⇒ Either         x+10=0          or         x-9 = 0

                         x = -10         or         x = 9

Now on taking any value of x satisfies given condition so –

Required Integer can be -10 or 9.

问题 6：求减 20 后等于该数倒数的 69 倍的整数。

解决方案：

Let the whole number is=x

So reciprocal of the number is = 1/x

now according to condition-

⇒ Number-20=69*(Reciprocal of the number)

⇒ x-20=69*(1/x);

⇒ x-(69/x)-20=0

by taking LCM-

⇒ (x²-69-20x)/x = 0

but denominator can’t be equal to 0  so-

⇒ (x²-20x-69)=0

⇒ x² – (23-3)x -69=0

⇒ x² – 23x + 3x -69=0

⇒ x(x-23)+3(x-23)=0

⇒(x-23)(x+3)=0

Either x+3=0      or      x-23=0

            x=-3       or        x=23

but x=-3 is not a whole number

so taking x=23, it is a whole number

⇒ So our answer is x=23

问题 7：找到两个连续的自然数，其乘积为 20。

解决方案：

Let the first number is = x

So next number is = x+1

Now according to condition-

⇒(first number)*(second consecutive number) = 20

⇒ x(x+1)=20

⇒ x² + x = 20

⇒ x² + x – 20=0

⇒ x² +(5-4)x – 20=0

⇒ x² + 5x – 4x – 20=0

⇒ x(x+5)-4(x+5)=0

⇒(x+5)(x-4)=0

⇒Either x+5=0      or     x-4=0

              x=-5        or      x=4

But x=-5 is not a natural number,

So taking x=4,

First number=4

and Second number=x+1

Second number=5

问题8：两个连续奇数正整数的平方和为394。求它们。

解决方案：

Let the first positive odd number is = x

so Second positive odd number is = x+2

Now according to the condition-

 ⇒(First number)²+(Second number)² = 394

⇒ x² + (x+2)² = 394

⇒ x² + x² + 2*x*2 + 4 = 394            [because (a+b)² = a² + 2*a*b + b²]

⇒ 2x² + 4x + 4 = 394

⇒  2x² + 4x – 390 = 0

Dividing by 2 –

⇒ x² + 2x – 195 = 0

⇒ x² + (15-13)x – 195 = 0

⇒ x² + 15x – 13x -195 = 0

⇒ x(x+15) – 13(x+15) = 0

⇒ (x+15)(x-13) = 0

Either x+15=0       or     x-13=0

               x=-15       or       x=13

but x=-15 is not a positive odd number

So on taking x=13

First positive odd number=13

and Second number = x+2

Second number = 15

问题 9：两个数之和是 8，它们的倒数之和也是 8 的 15 倍。求这些数。

解决方案：

Let first number is = x

and sum of numbers is given which is = 8 

So second number is = (8-x)

Now,

Reciprocal of first number is = 1/x

And reciprocal of second number is = 1/(8-x)

Given condition is-

⇒15[(1/x)+(1/(8-x))]=8

⇒15[((8-x)+x)/(x(8-x))]=8   [by taking LCM]

⇒15*[8-x+x]=8[x(8-x)]

⇒15*8 = 8[8x-x²]

Dividing by 8–

⇒ 15 = 8x-x²

⇒x²-8x+15=0

⇒x²-(5+3)x+15=0

⇒x²-5x-3x+15=0

⇒x(x-5)-3(x-5)=0

⇒(x-5)(x-3)=0

Either x-5=0      or       x-3=0

x=5       or          x=3

So numbers are 3,5.

问题 10：一个数与其正平方根之和为 6/25。找到号码。

解决方案：

Let the number is = x

So it’s square root is = √x

Now according to condition-

⇒x+√x=6/25

⇒x-6/25=-√x

Now squaring both sides-

⇒(x-6/25)²=(-√x)²

⇒x² – 2*x*(6/25) + (6/25)² = x      [because (a+b)² = a² + 2*a*b + b²]

⇒x²-(12/25)x+(36/625) = x         [because (6/25)²=36/625]

⇒ x² – (12/25)x -x +(36/625) = 0

⇒x² -[(12/25)+1]x + (36/625) = 0

⇒ x² -(37/25)x + (36/625) = 0

now for making factor-

⇒ x² -[(36/25)+(1/25)]x + (36/625) = 0     

⇒x² – (36/25)x – (1/25)x + (36/625) = 0  

⇒x[x-(36/25)] – (1/25)[x-(36/25)]=0

⇒[x-(1/25)][x-(36/25)]=0

Either    x-(1/25) = 0          or       x-(36/25)=0

                  x=(1/25)           or           x=(36/25)

but when x=(36/25)

then     (36/25)+√(36/25)=(36/25)+(6/5)

                                        = (36+30)/25

                                        = 66/25

So when x=36/25  , it doesn’t fulfill given condition.

So required number = 1/25.