Multiplication of the sum of two terms by the difference between them - Examples, Exercises and Solutions

(X+Y)×(XY)=X2Y2(X + Y)\times (X - Y) = X^2 - Y^2

This is one of the shortened multiplication formulas.

As can be seen, this formula can be used when there is a multiplication between the sum of two particular elements and the subtraction between the two elements.
Instead of presenting them as a multiplication of sum and subtraction, it can be written X2Y2X^2 - Y^2 and it expresses exactly the same thing. In the same way, if such an expression X2Y2X^2 - Y^2 representing the subtraction of two squared numbers is presented to you, you can write it like this: (X+Y)×(XY)(X + Y)\times (X - Y)
Pay attention: the formula works both in non-algebraic expressions and in expressions that combine unknowns and numbers.

Suggested Topics to Practice in Advance

  1. The formula for the sum of squares
  2. The formula for the difference of squares

Practice Multiplication of the sum of two terms by the difference between them

Exercise #1

Solve:

(2+x)(2x)=0 (2+x)(2-x)=0

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

4x2=0 4-x^2=0

We isolate the terms and extract the root:

4=x2 4=x^2

x=4 x=\sqrt{4}

x=±2 x=\pm2

Answer

±2

Exercise #2

(2x)23=6 (2x)^2-3=6

Video Solution

Step-by-Step Solution

We move the sections and equal to 0

4x236=0 4x^2-3-6=0

4x29=0 4x^2-9=0

We use the shortcut multiplication formula:

4(x294)=0 4(x^2-\frac{9}{4})=0

x2(32)2=0 x^2-(\frac{3}{2})^2=0

(x32)(x+32)=0 (x-\frac{3}{2})(x+\frac{3}{2})=0

x=±32 x=\pm\frac{3}{2}

Answer

±32 ±\frac{3}{2}

Exercise #3

Complete the following exercise:

(x+12)(x12)=0 (\sqrt{x}+\frac{1}{2})(\sqrt{x}-\frac{1}{2})=0

Video Solution

Answer

14 \frac{1}{4}

Exercise #4

Solve the exercise:

(x+3)(x3)+(x+1)(x1)=0 (x+3)(x-3)+(x+1)(x-1)=0

Video Solution

Answer

±5 ±\sqrt{5}

Exercise #5

Fill in the missing element to obtain a true expression:

(x+)(x)=x2121 (x+_—)\cdot(x-_—)=x^2-121

Video Solution

Answer

11

Exercise #1

Fill in the missing element to obtain a true expression:

(+3)(3)=x29 (_—+3)\cdot(_—-3)=x^2-9

Video Solution

Answer

x x

Exercise #2

(x+7)(x7)3=11x2 \frac{(x+7)(x-7)}{3}=-11-x^2

Video Solution

Answer

±2

Exercise #3

2x2328=x+42 \frac{2x^2-32}{8}=\frac{x+4}{2}

Video Solution

Answer

6

Exercise #4

Solve the following equation:

(x+8)(8x)+4(x3)(x+3)+5(6x2)=0 (x+8)(8-x)+4(x-3)(x+3)+5(6-x^2)=0

Video Solution

Answer

±29 ±\sqrt{29}

Exercise #5

Solve the following equation:

(x7)(x+7)=x2+7x+7 (x-\sqrt{7})(x+\sqrt{7})=x^2+7x+7

Video Solution

Answer

-2

Exercise #1

Fill in the missing element to obtain a true expression:

x264=(x)(+x) x^2-64=(x-_—)(_—+x)

Video Solution

Answer

8

Exercise #2

Fill in the missing element to obtain a true expression:

x249=(x)(x+) x^2-49=(x-_—)\cdot(x+_—)

Video Solution

Answer

7

Exercise #3

x26=(x)(x+) x^2-6=(x-_—)\cdot(x+_—)

Video Solution

Answer

6 \sqrt{6}

Exercise #4

Fill in the missing element to obtain a true expression:

x236=(x)(+x) x^2-36=(x-_—)\cdot(_—+x)

Video Solution

Answer

6

Exercise #5

Fill in the missing element to obtain a true expression:

2x2=2(x4)(x+4) 2x^2-_{_—}=2(x-4)\cdot(x+4)

Video Solution

Answer

32

Topics learned in later sections

  1. Abbreviated Multiplication Formulas