Examples with solutions for Square of Difference: Complete the missing numbers

Exercise #1

Fill in the blanks:

(x2y)2+(y2x)2=(?)×(x2+1)+2xy×(?)+y2×(?) (x^2-y)^2+(y^2-x)^2=(?)\times(x^2+1)+2xy\times(?)+y^2\times(?)

Step-by-Step Solution

To solve this problem, let's consider the expansion of the expressions:

First, expand each square:

  • (x2y)2=(x2)22(x2)y+y2(x^2 - y)^2 = (x^2)^2 - 2(x^2)y + y^2
  • (y2x)2=(y2)22(y2)x+x2(y^2 - x)^2 = (y^2)^2 - 2(y^2)x + x^2

Expanding these expressions, we get:

(x2y)2=x42x2y+y2(x^2 - y)^2 = x^4 - 2x^2y + y^2
(y2x)2=y42y2x+x2(y^2 - x)^2 = y^4 - 2y^2x + x^2

Adding the two expanded forms gives:

x42x2y+y2+y42y2x+x2=x4+y4+x2+y22x2y2y2xx^4 - 2x^2y + y^2 + y^4 - 2y^2x + x^2 = x^4 + y^4 + x^2 + y^2 - 2x^2y - 2y^2x

Now rearrange the terms in a form similar to the given expression:

The goal is to arrange the polynomial as: a(x2+1)+2xbcxy+c(y2+1)a(x^2 + 1) + 2xbcxy + c(y^2 + 1).

Since now we want: (x2y)2+(y2x)2=a(x2+1)+2xy((x+y))+c(y2+1)(x^2 - y)^2 + (y^2 - x)^2 = a(x^2 + 1) + 2xy(- (x+y)) + c(y^2 + 1).

Create matching forms:

  • The term for a(x2+1)a(x^2 + 1) will have a=x2a = x^2.
  • The term for bb in 2xy((x+y))2xy(- (x+y)) will have b=(x+y)b = -(x+y).
  • The term for c(y2+1)c(y^2 + 1) will have c=y2+1c = y^2 + 1.

Therefore, the correct values to fill in are (x2,(x+y),y2+1)(x^2,-(x+y),y^2+1).

Thus, our complete expression is:

(x2y)2+(y2x)2=x2(x2+1)+2xy((x+y))+y2(y2+1) (x^2 - y)^2 + (y^2 - x)^2 = x^2(x^2+1) + 2xy(-(x+y)) + y^2(y^2+1)

Hence, the answer is:

x2,(x+y),y2+1 x^2,-(x+y),y^2+1

Answer

x2,(x+y),y2+1 x^2,-(x+y),y^2+1

Exercise #2

Fill in the blanks:

x2+?+9=(x3)2 x^2+?+9=(x-3)^2

Video Solution

Answer

6x -6x

Exercise #3

Fill in the blanks:

(2x?)2=4x212x+? (2x-?)^2=4x^2-12x+\text{?}

Video Solution

Answer

3, 9 3,\text{ }9

Exercise #4

Fill in the blanks:

(x?)2=x2?+25 (x-?)^2=x^2-?+25

Video Solution

Answer

5, 10x 5,\text{ }10x

Exercise #5

Fill in the blanks:

(?×x?)2=9x224x+16 (?\times x-?)^2=9x^2-24x+16

Video Solution

Answer

3, 4 3,\text{ }4

Exercise #6

Fill in the blanks:

(?×x1)2+(?×x2)2=5x28x+5 (?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5

Video Solution

Answer

2, 1 2,\text{ }1