Combining Short Multiplication Formulas - Examples, Exercises and Solutions

Meet the square of binomials formulas:

Product of the sum of two terms and their difference

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

Click here to read more about Multiplication of the sum of two elements by the difference between them!

The difference of squares formula

$(X - Y)^2=X^2 - 2XY + Y^2$

Click here to read more about the formula for the difference of squares

The Square of a Sum Formula

$(X + Y)^2=X^2+ 2XY + Y^2$

Click here to read more about the formula for the Sum of Squares

Formulas relating to two expressions cubed

$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$

$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

Click here to read more about Formulas for Cubic Expressions

Example

Practice an exercise that combines all the shortened multiplication formulas together:
$x^2+(5+x)(5-x)+x^2-6x+9-(6^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3)=(2+x)^2+(6+x)^3$

Let's start from the beginning of the exercise. Observe that the expression
$(5+x)(5-x)=25-x^2$
matches the product of the sum of th two terms and their difference
$(X + Y)\cdot(X - Y) = X^2 - Y^2$
Let's proceed to the expression $x^2-6x+9$
We notice that it matches the difference of squares formula
$(X - Y)^2=X^2 - 2XY + Y^2$
However let's avoid touching it at this stage.
Let's continue to $6^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3$
and we can see that it perfectly matches the formula for two terms cubed
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
which means
$6^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3=(6+x)^3$

Let's continue to the second side and observe that the expression $(2+x)^2$ matches the formula for sum of squares
$(X + Y)^2=X^2+ 2XY + Y^2$
Therefore
$(2+x)^2=4+4x+x^2$

Now let's insert the data:
$x^2+25-x^2+x^2-6x+9-(6+x)^3=4+4x+x^2+(6+x)^3$
Let's reduce the terms and solve as follows:
$-6x+34=4x+4$
Move the terms to opposite sides:
$8x=30$
$x=3.75$