Equivalent Expressions – SAT Math

📖 What Are Equivalent Expressions?

Equivalent expressions are expressions that have the same value for all values of the variable(s). They may look different but simplify to the same thing.

\(2(x + 3) = 2x + 6\)

Both expressions are equivalent — they have the same value for any \(x\).

📌 Key Concept

Equivalent expressions are created using algebraic properties:
• Distributive Property
• Combining Like Terms
• Factoring
• Exponent Rules

⚡ Key Rules for Equivalent Expressions

Distributive Property

\(a(b + c) = ab + ac\)

Multiply each term inside parentheses by the factor outside.

Combining Like Terms

\(3x + 2x = 5x\)

Add or subtract terms with the same variable and exponent.

Factoring (GCF)

\(6x + 9 = 3(2x + 3)\)

Find the greatest common factor and factor it out.

Difference of Squares

\(a^2 - b^2 = (a - b)(a + b)\)

Two perfect squares subtracted.

Perfect Square Trinomial

\((a \pm b)^2 = a^2 \pm 2ab + b^2\)

Square of a binomial.

Product of Binomials

\((x + a)(x + b) = x^2 + (a+b)x + ab\)

FOIL method: First, Outer, Inner, Last.

📖 Detailed Rules

📌 RULE 1: Distributive Property

\[ a(b + c) = ab + ac \]
\[ a(b - c) = ab - ac \]

Example: \(3(2x + 4) = 3(2x) + 3(4) = 6x + 12\)

📌 RULE 2: Combining Like Terms

Only terms with the same variable and same exponent can be combined.

Example: \(3x^2 + 5x - 2x^2 + 3x = (3x^2 - 2x^2) + (5x + 3x) = x^2 + 8x\)

📌 RULE 3: Factoring (Greatest Common Factor)

Find the largest factor common to all terms.

Example: \(6x^2 + 9x = 3x(2x + 3)\)
(GCF = \(3x\))

📌 RULE 4: Difference of Squares

\[ a^2 - b^2 = (a - b)(a + b) \]

Example: \(x^2 - 16 = (x - 4)(x + 4)\)

📌 RULE 5: Perfect Square Trinomial

\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

Example: \((x + 3)^2 = x^2 + 6x + 9\)

📌 RULE 6: Product of Binomials (FOIL)

\[ (x + a)(x + b) = x^2 + (a+b)x + ab \]

Example: \((x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10\)

💡 Strategy — Equivalent Expressions

1. Simplify by combining like terms and using the distributive property.
2. Factor by finding the GCF or using special patterns (difference of squares, perfect square trinomials).
3. Expand using FOIL or the distributive property.
4. Always check if the expressions are equal by testing a value or simplifying both sides.

📝 Solved Examples

Study these examples carefully. Each shows the step-by-step solution process.

📝 SOLVED EXAMPLE 1 — Simplifying

Simplify: \(3(2x + 4) - 2(x - 1)\)

Step 1: Distribute: \(6x + 12 - 2x + 2\)

Step 2: Combine like terms: \((6x - 2x) + (12 + 2)\)

Step 3: \(\color{var(--math)}{4x + 14}\)

✅ \(4x + 14\)

💡 Tip: Be careful with the negative sign! \( -2(x - 1) = -2x + 2\)

📝 SOLVED EXAMPLE 2 — Factoring

Factor completely: \(8x^2 + 12x\)

Step 1: Find the GCF: \(4x\) (both terms divisible by \(4x\))

Step 2: Factor out: \(\color{var(--math)}{4x(2x + 3)}\)

✅ \(4x(2x + 3)\)

💡 Tip: Check: \(4x(2x + 3) = 8x^2 + 12x\) ✓

📝 SOLVED EXAMPLE 3 — Difference of Squares

Factor: \(x^2 - 25\)

Step 1: Recognize: \(x^2 - 25 = x^2 - 5^2\)

Step 2: Use formula: \(a^2 - b^2 = (a - b)(a + b)\)

Step 3: \(\color{var(--math)}{(x - 5)(x + 5)}\)

✅ \((x - 5)(x + 5)\)

💡 Tip: Check: \((x - 5)(x + 5) = x^2 - 25\) ✓

📝 SOLVED EXAMPLE 4 — Expanding

Expand: \((2x + 3)^2\)

Step 1: Use \((a + b)^2 = a^2 + 2ab + b^2\)

Step 2: \(a = 2x\), \(b = 3\)

Step 3: \((2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9\)

✅ \(4x^2 + 12x + 9\)

💡 Tip: Check: \((2x + 3)(2x + 3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9\) ✓

📝 SOLVED EXAMPLE 5 — Factoring Trinomial

Factor: \(x^2 + 5x + 6\)

Step 1: Find two numbers that multiply to 6 and add to 5: \(2\) and \(3\)

Step 2: \(\color{var(--math)}{(x + 2)(x + 3)}\)

✅ \((x + 2)(x + 3)\)

💡 Tip: Check: \((x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6\) ✓

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

Simplify: \(2(3x + 5) - 4(x - 2)\)

A) \(2x + 18\)

B) \(2x + 18\)

C) \(6x + 7\)

D) \(10x + 3\)

✓ Answer: B

\(2(3x + 5) - 4(x - 2) = 6x + 10 - 4x + 8 = 2x + 18\)

📝 Solution: \(6x + 10 - 4x + 8 = 2x + 18\)

Question 2

Factor: \(6x^2 - 15x\)

A) \(3x(2x - 5)\)

B) \(3x(2x + 5)\)

C) \(3x(2x - 5)\)

D) \(6x(x - 2.5)\)

✓ Answer: C

GCF is \(3x\): \(6x^2 - 15x = 3x(2x - 5)\)

📝 Solution: \(3x(2x - 5)\)

Question 3

Factor: \(x^2 - 36\)

A) \((x - 6)(x + 6)\)

B) \((x - 6)(x - 6)\)

C) \((x + 6)(x + 6)\)

D) \(x(x - 36)\)

✓ Answer: A

\(x^2 - 36 = x^2 - 6^2 = (x - 6)(x + 6)\)

📝 Solution: \((x - 6)(x + 6)\)

Question 4

Expand: \((x + 4)^2\)

A) \(x^2 + 4x + 16\)

B) \(x^2 + 8x + 16\)

C) \(x^2 + 16x + 16\)

D) \(x^2 + 8\)

✓ Answer: B

\((x + 4)^2 = x^2 + 2(4)x + 4^2 = x^2 + 8x + 16\)

📝 Solution: \(x^2 + 8x + 16\)

Question 5

Factor: \(x^2 + 7x + 10\)

A) \((x + 2)(x + 5)\)

B) \((x + 1)(x + 10)\)

C) \((x - 2)(x + 5)\)

D) \((x + 2)(x + 5)\)

✓ Answer: D

Need two numbers that multiply to 10 and add to 7: 2 and 5 → \((x + 2)(x + 5)\)

📝 Solution: \((x + 2)(x + 5)\)

Question 6

Simplify: \(4(2x - 3) + 5(3x + 2)\)

A) \(23x - 2\)

B) \(23x - 2\)

C) \(7x - 2\)

D) \(23x - 22\)

✓ Answer: B

\(4(2x - 3) + 5(3x + 2) = 8x - 12 + 15x + 10 = 23x - 2\)

📝 Solution: \(8x - 12 + 15x + 10 = 23x - 2\)

Question 7

Factor: \(9x^2 - 16\)

A) \((3x - 4)(3x + 4)\)

B) \((3x - 4)(3x - 4)\)

C) \((9x - 4)(x + 4)\)

D) \(x(9x - 16)\)

✓ Answer: A

\(9x^2 - 16 = (3x)^2 - 4^2 = (3x - 4)(3x + 4)\)

📝 Solution: \((3x - 4)(3x + 4)\)

Question 8

Expand: \((2x - 3)^2\)

A) \(4x^2 - 12x + 9\)

B) \(4x^2 - 6x + 9\)

C) \(4x^2 - 12x + 9\)

D) \(4x^2 - 9\)

✓ Answer: C

\((2x - 3)^2 = (2x)^2 - 2(2x)(3) + 9 = 4x^2 - 12x + 9\)

📝 Solution: \(4x^2 - 12x + 9\)

Question 9

Factor: \(x^2 + 2x - 8\)

A) \((x + 4)(x - 2)\)

B) \((x - 2)(x + 4)\)

C) \((x - 4)(x + 2)\)

D) \((x + 1)(x - 8)\)

✓ Answer: B

Need two numbers that multiply to -8 and add to 2: 4 and -2 → \((x + 4)(x - 2)\)

📝 Solution: \((x + 4)(x - 2)\)

Question 10

Simplify: \(3(x^2 + 2x) - 2(x^2 - 4x + 3)\)

A) \(x^2 + 14x - 6\)

B) \(x^2 + 14x - 6\)

C) \(x^2 + 2x - 6\)

D) \(x^2 + 6x - 6\)

✓ Answer: B

\(3(x^2 + 2x) - 2(x^2 - 4x + 3) = 3x^2 + 6x - 2x^2 + 8x - 6 = x^2 + 14x - 6\)

📝 Solution: \(3x^2 + 6x - 2x^2 + 8x - 6 = x^2 + 14x - 6\)

🎉 WELL DONE!

You've completed the Equivalent Expressions lesson. You now know how to simplify, factor, expand, and work with equivalent forms of expressions.

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