Digital SAT Maths -Unit 2 - 2.1 Equivalent Expressions- Study Notes- New Syllabus
Digital SAT Maths -Unit 2 – 2.1 Equivalent Expressions- Study Notes- New syllabus
Digital SAT Maths -Unit 2 – 2.1 Equivalent Expressions- Study Notes – per latest Syllabus.
Key Concepts:
Expanding (distributive law)
Factorization (common, grouping)
Rewriting expressions for purpose
Expanding (Distributive Law)
Two expressions are equivalent if they have the same value for every value of the variable.
One of the most important algebra skills on the DIGITAL SAT is expanding expressions using the distributive law.
Distributive Law
\( a(b+c)=ab+ac \)
You multiply the outside term by every term inside the bracket.
Important Rules
- Multiply to every term inside parentheses
- Watch negative signs carefully
- Combine like terms at the end
Double Brackets (FOIL Method)
For two binomials:
\( (a+b)(c+d)=ac+ad+bc+bd \)
(sometimes remembered as FOIL: First, Outer, Inner, Last)
DIGITAL SAT Tip
Many SAT questions do not ask you to “expand.” Instead they ask which expression is equivalent. That is still a distributive law question.
Example 1:
Expand \( 3(2x+5) \).
▶️ Answer/Explanation
\( 3\cdot2x + 3\cdot5 \)
\( =6x+15 \)
Answer: \( 6x+15 \)
Example 2:
Expand \( -2(x-4) \).
▶️ Answer/Explanation
\( -2\cdot x + (-2)(-4) \)
\( =-2x+8 \)
Answer: \( -2x+8 \)
Example 3:
Which expression is equivalent to \( (x+3)(x+2) \)?
▶️ Answer/Explanation
\( x(x+2)+3(x+2) \)
\( =x^2+2x+3x+6 \)
\( =x^2+5x+6 \)
Answer: \( x^2+5x+6 \)
Factorization (Common Factor & Grouping)
Factorization is the reverse of expanding. Instead of multiplying brackets, we rewrite an expression as a product.
This is extremely important on the DIGITAL SAT because questions often ask you to:
- simplify expressions
- solve equations
- identify a hidden meaning (like zeros or intercepts)
1. Common Factor
Take out the greatest common factor (GCF) from all terms.
\( ab + ac = a(b+c) \)
Steps
- Find the common number or variable
- Factor it outside brackets
- Divide each term by it
2. Factorization by Grouping
Used when there are four terms.
\( ax+ay+bx+by \)
Group them in pairs:
\( a(x+y)+b(x+y) \)
\( (a+b)(x+y) \)
DIGITAL SAT Tip
Many SAT problems hide a common factor. If every term has an \( x \), factor it first.
Example 1 (Common Factor):
Factor \( 6x + 12 \).
▶️ Answer/Explanation
GCF = 6
\( 6(x+2) \)
Answer: \( 6(x+2) \)
Example 2 (Variable Common Factor):
Factor \( 8x^2 – 4x \).
▶️ Answer/Explanation
Common factor = \( 4x \)
\( 4x(2x-1) \)
Answer: \( 4x(2x-1) \)
Example 3 (Grouping):
Factor \( 3x + 3y + 2x + 2y \).
▶️ Answer/Explanation
Group terms:
\( (3x+3y)+(2x+2y) \)
Factor each group:
\( 3(x+y)+2(x+y) \)
\( (3+2)(x+y) \)
\( 5(x+y) \)
Answer: \( 5(x+y) \)
Rewriting Expressions for a Purpose
On the DIGITAL SAT, you are often not asked to simply simplify. Instead, you must rewrite an expression to reveal useful information.
The expression stays mathematically equivalent, but the new form makes something easier to see.
Why Rewrite?
- to identify a constant
- to find intercepts
- to interpret meaning in context
- to make mental calculation easier
Common Techniques
1. Factoring to Find Zeros
\( x^2+5x+6=(x+2)(x+3) \)
Now you can easily see the values that make the expression 0.
2. Completing the Square (Reveal Minimum/Maximum)
\( x^2+6x+5=(x+3)^2-4 \)
This reveals the vertex of a parabola.
3. Rearranging Linear Expressions
Example:
\( 5x+20=5(x+4) \)
This helps identify the meaning of the constant.
DIGITAL SAT Tip
If the problem asks “What does this constant represent?”, you almost always need to rewrite the expression.
Example 1 (Reveal Zeros):
Rewrite \( x^2-7x+10 \) to identify the solutions of \( x^2-7x+10=0 \).
▶️ Answer/Explanation
\( x^2-7x+10=(x-5)(x-2) \)
\( x=5 \) or \( x=2 \)
Conclusion: The solutions are 2 and 5.
Example 2 (Interpret a Constant):
A gym membership cost is given by \( C(x)=25x+50 \), where \( x \) is the number of months. Rewrite and interpret the constant.
▶️ Answer/Explanation
Already in slope-intercept form:
\( C(x)=25x+50 \)
The constant 50 is the cost when \( x=0 \).
Conclusion: \$50 is the initial signup fee.
Example 3 (Complete the Square):
Rewrite \( x^2+4x+1 \) in the form \( (x+a)^2+b \).
▶️ Answer/Explanation
\( x^2+4x+1=(x+2)^2-3 \)
Answer: \( (x+2)^2-3 \)