Positive Distribution: $a(b + c) = ab + ac$
Negative Distribution: $a(b – c) = ab – ac$
Right-side Distribution (Addition): $(a + b)c = ac + bc$
Right-side Distribution (Subtraction): $(a – b)c = ac – bc$
When multiplying two binomials, every term in the first set of parentheses must be multiplied by every term in the second:
$(a + b)(c + d) = ac + ad + bc + bd$
$(a + b)(c – d) = ac – ad + bc – bd$
$(a – b)(c + d) = ac + ad – bc – bd$
$(a – b)(c – d) = ac – ad – bc + bd$
The document highlights the identity for the difference of squares:
The proof involves expanding the binomials:
$(a + b)(a – b) = a^2 – ab + ab – b^2$
Since $-ab$ and $+ab$ cancel each other out, we are left with:
$= a^2 – b^2$
The identity can be used to simplify products or evaluate large numbers quickly:
| Problem Type | Example Calculation | Result |
| Variables | $(2x + 3y)(2x – 3y) = (2x)^2 – (3y)^2$ | $4x^2 – 9y^2$ |
| Fractions | $(x + \frac{1}{x})(x – \frac{1}{x}) = (x)^2 – (\frac{1}{x})^2$ | $x^2 – \frac{1}{x^2}$ |
| Arithmetic | $297 \times 303 = (300 – 3)(300 + 3) = 300^2 – 3^2$ | $89991$ |
| Squares | $51^2 – 49^2 = (51 + 49)(51 – 49) = 100(2)$ | $200$ |
When multiplying a single term into a group, you “distribute” it to every item inside the parentheses:
$a(b + c) = ab + ac$
$a(b – c) = ab – ac$
$(a + b)c = ac + bc$
$(a – b)c = ac – bc$
When multiplying two binomials, every term in the first set must be multiplied by every term in the second.
$(a + b)(c + d) = ac + ad + bc + bd$
$(a – b)(c – d) = ac – ad – bc + bd$
Identities are equations that are true for all values of the variables. Here are the three most essential ones:
Identity: $(a + b)(a – b) = a^2 – b^2$
The Proof: By expanding the left side, we get $a^2 – ab + ab – b^2$. The middle terms cancel out, leaving $a^2 – b^2$.
Example: $(2x + 3y)(2x – 3y) = (2x)^2 – (3y)^2 = 4x^2 – 9y^2$.
Identity: $(a + b)^2 = a^2 + b^2 + 2ab$
The Proof: $(a + b)(a + b) = a^2 + ab + ab + b^2$, which simplifies to $a^2 + 2ab + b^2$.
Example: $(102)^2 = (100 + 2)^2 = 100^2 + 2^2 + 2(100)(2) = 10,404$.
Identity: $(a – b)^2 = a^2 + b^2 – 2ab$
The Proof: Expanding $(a – b)(a – b)$ gives $a^2 – ab – ab + b^2$, which simplifies to $a^2 – 2ab + b^2$.
Example: $(4x – 7y)^2 = (4x)^2 + (7y)^2 – 2(4x)(7y) = 16x^2 + 49y^2 – 56xy$.
| Identity Name | Formula | Use Case Example |
| Difference of Squares | $(a+b)(a-b) = a^2 – b^2$ | Mental math like $297 \times 303$ |
| Square of Sum | $(a+b)^2 = a^2 + 2ab + b^2$ | Squaring numbers like $102^2$ |
| Square of Difference | $(a-b)^2 = a^2 – 2ab + b^2$ | Expanding $(4x – 7y)^2$ |
When you multiply a single term into a group, it must be applied to every item inside the parentheses:
$a(b + c) = ab + ac$
$a(b – c) = ab – ac$
$(a + b)c = ac + bc$
$(a – b)c = ac – bc$
When multiplying two binomials, follow the rule that every combination must be multiplied:
$(a + b)(c + d) = ac + ad + bc + bd$
$(a – b)(c – d) = ac – ad – bc + bd$
Identities are formulas that are always true, regardless of the values you plug in. Here are the four most vital identities and how they work.
Formula: $(a + b)(a – b) = a^2 – b^2$
The Proof: Expanding $(a + b)(a – b)$ gives $a^2 – ab + ab – b^2$. The middle terms cancel out, leaving just $a^2 – b^2$.
Practical Use: This is perfect for mental math. For example, $297 \times 303$ can be rewritten as $(300 – 3)(300 + 3)$, which simplifies to $300^2 – 3^2$, or $90,000 – 9 = 89,991$.
Formula: $(a + b)^2 = a^2 + b^2 + 2ab$
The Proof: $(a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$.
Example: To find $(102)^2$, think of it as $(100 + 2)^2$. This gives $100^2 + 2^2 + 2(100)(2)$, resulting in $10,000 + 4 + 400 = 10,404$.
Formula: $(a – b)^2 = a^2 + b^2 – 2ab$
Example: $(4x – 7y)^2 = (4x)^2 + (7y)^2 – 2(4x)(7y)$, which simplifies to $16x^2 + 49y^2 – 56xy$.
Formula: $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$
The Proof: If you add the expansion of $(a + b)^2$ to the expansion of $(a – b)^2$, the $+2ab$ and $-2ab$ terms cancel out, leaving $2a^2 + 2b^2$.
Simplification Example: To simplify $(2x + 3y)^2 + (2x – 3y)^2$, you can immediately write $2[(2x)^2 + (3y)^2]$, which is $2[4x^2 + 9y^2]$.
| Identity Name | Formula | Best Used For… |
| Difference of Squares | $(a+b)(a-b) = a^2 – b^2$ | Rapidly multiplying numbers near a common base. |
| Square of Sum | $(a+b)^2 = a^2 + 2ab + b^2$ | Squaring numbers just above a round number. |
| Square of Difference | $(a-b)^2 = a^2 – 2ab + b^2$ | Squaring numbers just below a round number. |
Positive Distribution: $a(b + c) = ab + ac$
Negative Distribution: $a(b – c) = ab – ac$
Right-side Distribution: $(a + b)c = ac + bc$ or $(a – b)c = ac – bc$
When multiplying two binomials, follow the rule that every combination must be multiplied:
$(a + b)(c + d) = ac + ad + bc + bd$
$(a – b)(c – d) = ac – ad – bc + bd$
Identities are equations that are true for all values of the variables. Here are the most essential identities, their proofs, and how to apply them.
Identity: $(a + b)(a – b) = a^2 – b^2$
The Proof: Expanding $(a + b)(a – b)$ gives $a^2 – ab + ab – b^2$. The middle terms cancel, leaving $a^2 – b^2$.
Mental Math Example: To evaluate $297 \times 303$, rewrite it as $(300 – 3)(300 + 3)$. This becomes $300^2 – 3^2 = 90000 – 9 = 89991$.
Identity: $(a + b)^2 = a^2 + b^2 + 2ab$
The Proof: $(a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$.
Example: $(102)^2 = (100 + 2)^2 = 100^2 + 2^2 + 2(100)(2) = 10000 + 4 + 400 = 10404$.
Identity: $(a – b)^2 = a^2 + b^2 – 2ab$
Example: $(4x – 7y)^2 = (4x)^2 + (7y)^2 – 2(4x)(7y) = 16x^2 + 49y^2 – 56xy$.
These identities show how the square of a sum and the square of a difference relate to one another.
| Identity Formula | Application Example |
| $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$ | Simplifying $(2x+3y)^2 + (2x-3y)^2 = 2[4x^2 + 9y^2]$ |
| $(a + b)^2 – (a – b)^2 = 4ab$ | Simplifying $(x+2y)^2 – (x-2y)^2 = 4(x)(2y) = 8xy$ |
| $(a + b)^2 = (a – b)^2 + 4ab$ | Showing $(5x-7y)^2 + 140xy = (5x+7y)^2$ |
Basic Addition: $a(b+c) = ab + ac$
Basic Subtraction: $a(b-c) = ab – ac$
Right-side Distribution: $(a+b)c = ac + bc$ and $(a-b)c = ac – bc$
When multiplying two binomials, use the “FOIL” pattern to ensure all terms interact:
$(a+b)(c+d) = ac + ad + bc + bd$
$(a-b)(c-d) = ac – ad – bc + bd$
These identities are equations that remain true for any values of $a$ and $b$.
Identity: $(a+b)(a-b) = a^2 – b^2$
The Proof: Expanding the left side gives $a^2 – ab + ab – b^2$. The middle terms cancel out, leaving $a^2 – b^2$.
Mental Math Example: To evaluate $297 \times 303$, rewrite it as $(300-3)(300+3)$. This becomes $300^2 – 3^2 = 90000 – 9 = 89991$.
Identity: $(a+b)^2 = a^2 + b^2 + 2ab$
Example: $(102)^2$ can be solved as $(100+2)^2 = 100^2 + 2^2 + 2(100)(2) = 10000 + 4 + 400 = 10404$.
Identity: $(a-b)^2 = a^2 + b^2 – 2ab$
Example: $998^2$ can be solved as $(1000-2)^2 = 1000^2 + 2^2 – 2(1000)(2) = 1000000 + 4 – 4000 = 996004$.
These formulas help you move quickly between different squared forms.
| Identity | Simplified Result | Application Example |
| Sum of Squares | $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$ | $(2x+3y)^2 + (2x-3y)^2 = 2[4x^2 + 9y^2]$ |
| Difference of Squares | $(a+b)^2 – (a-b)^2 = 4ab$ | $(x+2y)^2 – (x-2y)^2 = 4(x)(2y) = 8xy$ |
| Converting to Sum | $(a-b)^2 + 4ab = (a+b)^2$ | Showing $(5x-7y)^2 + 140xy = (5x+7y)^2$ |
| Converting to Difference | $(a+b)^2 – 4ab = (a-b)^2$ | $(2x+3y)^2 – 24xy = (2x-3y)^2$ |
You can derive the sum of two squares from either the sum or difference identity:
$a^2 + b^2 = (a+b)^2 – 2ab$
$a^2 + b^2 = (a-b)^2 + 2ab$