Chapter 12: Identities - Your Secret Shortcut in Algebra!

Hey everyone! Welcome to the chapter on Identities. Think of identities as special "cheat codes" or shortcuts in mathematics. They are super useful tools that can help you solve problems much faster and with fewer steps. Once you learn them, you'll feel like an algebra superstar!

In this chapter, we'll learn:

1. What an identity is and how it's different from an equation.
2. How to use identities to expand algebraic expressions quickly.
3. How to use identities to factorise polynomials (which is like expanding in reverse!).


What's the Big Idea? Equations vs. Identities

You've worked with equations before, but what makes an identity different? It's a simple but very important difference.

Equations are true for SOME values

Think of an equation like a key that only opens one specific lock.

For example: The equation $$x + 3 = 8$$ is only true when x = 5. If you try any other number for x, it won't work.

Identities are true for ALL values

An identity is like a master key that opens every single lock! No matter what value you substitute for the variable, the statement will always be true.

For example: $$2(x + 1) = 2x + 2$$ Let's test it:

  • If x = 3: 2(3 + 1) = 2(4) = 8. And 2(3) + 2 = 6 + 2 = 8. It works!
  • If x = 10: 2(10 + 1) = 2(11) = 22. And 2(10) + 2 = 20 + 2 = 22. It works!
  • If x = -1: 2(-1 + 1) = 2(0) = 0. And 2(-1) + 2 = -2 + 2 = 0. It works!

Because it's true for any value of x, it's an identity. We often use the symbol '≡' instead of '=' for identities, but '=' is also commonly used.


How to Prove an Identity

To prove that an equation is an identity, your mission is to show that the Left Hand Side (L.H.S.) is exactly the same as the Right Hand Side (R.H.S.).

The Golden Rule: Work on one side only (usually the more complicated-looking side) and simplify it step-by-step until it looks identical to the other side.

Example: Prove that (x + 1)² - 1 = x(x + 2) is an identity.

Let's start with the L.H.S. because it looks more complicated.
L.H.S. = $$(x + 1)² - 1$$
Step 1: Expand $$(x + 1)²$$ which is $$(x + 1)(x + 1)$$ $$(x + 1)(x + 1) = x² + x + x + 1 = x² + 2x + 1$$
Step 2: Now put it back into the L.H.S. expression. $$L.H.S. = (x² + 2x + 1) - 1$$
Step 3: Simplify. $$L.H.S. = x² + 2x$$
Step 4: Now, let's look at the R.H.S. R.H.S. = $$x(x + 2) = x² + 2x$$
Since L.H.S. = R.H.S. ($$x² + 2x$$), we have proved it is an identity!

Key Takeaway

An equation is true for specific values, while an identity is true for all possible values. To prove an identity, simplify one side until it matches the other side.


Using Identities to Expand Expressions: The Fast Way!

Remember expanding brackets like $$(x+2)(x+3)$$? Identities are shortcuts for special cases of expansion. Let's learn the three most important ones!

Identity 1: The Difference of Two Squares

This identity is for when you multiply two brackets that are almost the same, but one has a 'plus' and the other has a 'minus'.

The Formula: $$(a - b)(a + b) = a² - b²$$

Memory Aid: Just remember "first thing squared, minus the second thing squared".

Example: Expand (x - 5)(x + 5).
Don't worry about the long way! Just use the identity.

Step 1: Identify 'a' and 'b'. Here, a = x and b = 5.
Step 2: Apply the formula $$a² - b²$$.
Step 3: Substitute your values in: $$x² - 5²$$
Answer: $$x² - 25$$. That's it! So much quicker!

Identity 2: The Perfect Square (The 'Plus' Version)

This is for when you square a bracket with a plus sign inside, like $$(a+b)²$$.

The Formula: $$(a + b)² = a² + 2ab + b²$$

Memory Aid: "Square the first, plus twice the product, plus square the last."

Example: Expand (y + 4)².

Step 1: Identify 'a' and 'b'. Here, a = y and b = 4.
Step 2: Apply the formula $$a² + 2ab + b²$$.
Step 3: Substitute your values: $$y² + 2(y)(4) + 4²$$
Answer: $$y² + 8y + 16$$.

Common Mistake Alert!
A very common mistake is to think that $$(a + b)² = a² + b²$$. This is WRONG! Don't forget the middle term, $$2ab$$! Think of it like building a house: you need the foundation ($$a²$$), the roof ($$b²$$), and all the walls in between ($$2ab$$). You can't have a house with just a floor and a roof!

Identity 3: The Perfect Square (The 'Minus' Version)

This is for squaring a bracket with a minus sign, like $$(a-b)²$$. It's very similar to the plus version.

The Formula: $$(a - b)² = a² - 2ab + b²$$

Notice that only the middle term is negative. The last term ($$b²$$) is still positive, because squaring a negative number gives a positive result!

Example: Expand (2x - 3)².

Step 1: Identify 'a' and 'b'. Here, a = 2x and b = 3.
Step 2: Apply the formula $$a² - 2ab + b²$$.
Step 3: Substitute carefully: $$(2x)² - 2(2x)(3) + 3²$$
Answer: $$4x² - 12x + 9$$.

Key Takeaway

Memorise these three expansion identities to save time:

  • $$(a - b)(a + b) = a² - b²$$
  • $$(a + b)² = a² + 2ab + b²$$
  • $$(a - b)² = a² - 2ab + b²$$

Using Identities to Factorise: Working Backwards!

Factorising is the reverse of expanding. We start with the answer (like $$x² - 25$$) and work backwards to find the original brackets. The same three identities are our tools!

Don't worry if this seems tricky at first. The key is learning to spot the patterns!

Factorising the Difference of Two Squares

If you see two perfect square terms separated by a minus sign, you can use this identity!

The Formula: $$a² - b² = (a - b)(a + b)$$

Example: Factorise y² - 49.

Step 1: Spot the pattern. Is it a difference of two squares? Yes! $$y²$$ is a square, $$49$$ is a square ($$7²$$), and they are separated by a minus.
Step 2: Identify 'a' and 'b'. What was squared to get each term? $$a² = y² \rightarrow a = y$$ $$b² = 49 \rightarrow b = 7$$
Step 3: Apply the formula $$(a - b)(a + b)$$.
Answer: $$(y - 7)(y + 7)$$.

Factorising Perfect Square Trinomials

A "trinomial" is just an expression with three terms. If you see three terms, check if it fits the perfect square pattern.

The Formulas:

  • $$a² + 2ab + b² = (a + b)²$$
  • $$a² - 2ab + b² = (a - b)²$$

How to spot the pattern: A 3-Step Check
1. Is the first term a perfect square? ($$a²$$)
2. Is the last term a perfect square? ($$b²$$)
3. Is the middle term $$2 \times a \times b$$? (Check the sign too!)

Example: Factorise x² + 14x + 49.

Step 1: Check the first and last terms.
First term: $$x²$$. Yes, this is $$(x)²$$. So, let's guess a = x.
Last term: $$49$$. Yes, this is $$7²$$. So, let's guess b = 7.

Step 2: Check the middle term.
The formula needs the middle term to be $$2ab$$. Let's check: $$2 \times (x) \times (7) = 14x$$. It matches perfectly!
Step 3: Decide the sign.
The middle term is $$+14x$$, so we use the plus version: $$(a + b)²$$.
Answer: $$(x + 7)²$$.

Key Takeaway

To factorise, look for visual clues!

  • Two terms with a minus? Check for $$a² - b²$$.
  • Three terms? Check if the first and last terms are squares, then check if the middle term is $$2ab$$.

You've Mastered Identities!

Great job! You've learned the three fundamental identities that will be your best friends in algebra. They might seem like a lot to remember at first, but with practice, you'll start spotting them everywhere.

Keep practicing using them for both expanding and factorising, and soon it will become second nature. You've got this!