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

$(X + Y)\times (X - Y) = X^2 - Y^2$

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 $X^2 - Y^2$ and it expresses exactly the same thing. In the same way, if such an expression $X^2 - Y^2$ representing the subtraction of two squared numbers is presented to you, you can write it like this: $(X + Y)\times (X - Y)$
Pay attention: the formula works both in non-algebraic expressions and in expressions that combine unknowns and numbers.

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

### Exercise #1

Solve:

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

### Step-by-Step Solution

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

±2

### Exercise #2

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

### Step-by-Step Solution

We move the sections and equal to 0

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

$4x^2-9=0$

We use the shortcut multiplication formula:

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

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

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

$x=\pm\frac{3}{2}$

$±\frac{3}{2}$

### Exercise #3

Complete the following exercise:

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

### Video Solution

$\frac{1}{4}$

### Exercise #4

Solve the exercise:

$(x+3)(x-3)+(x+1)(x-1)=0$

### Video Solution

$±\sqrt{5}$

### Exercise #5

Fill in the missing element to obtain a true expression:

$(x+_—)\cdot(x-_—)=x^2-121$

11

### Exercise #1

Fill in the missing element to obtain a true expression:

$(_—+3)\cdot(_—-3)=x^2-9$

### Video Solution

$x$

### Exercise #2

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

±2

### Exercise #3

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

6

### Exercise #4

Solve the following equation:

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

### Video Solution

$±\sqrt{29}$

### Exercise #5

Solve the following equation:

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

-2

### Exercise #1

Fill in the missing element to obtain a true expression:

$x^2-64=(x-_—)(_—+x)$

8

### Exercise #2

Fill in the missing element to obtain a true expression:

$x^2-49=(x-_—)\cdot(x+_—)$

7

### Exercise #3

$x^2-6=(x-_—)\cdot(x+_—)$

### Video Solution

$\sqrt{6}$

### Exercise #4

Fill in the missing element to obtain a true expression:

$x^2-36=(x-_—)\cdot(_—+x)$

6

### Exercise #5

Fill in the missing element to obtain a true expression:

$2x^2-_{_—}=2(x-4)\cdot(x+4)$