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Math foundations
SAT equivalent expressions: expand, factor, and check with a number
See multiplication as area, factor trinomials by finding two numbers, recognize special patterns at a glance, and check equivalence by substituting a number.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- An area model turns (x + 3)(x + 2) into four rectangles: x² + 2x + 3x + 6.
- To factor x² + bx + c, find two numbers that multiply to c and add to b.
- Two expressions are equivalent if they agree for every x; substituting a number such as x = 2 quickly checks a choice.
In this article
Multiplication as area
To expand \(\displaystyle (x + 3)(x + 2)\), picture a rectangle with side lengths \(\displaystyle x + 3\) and \(\displaystyle x + 2\). Split each side into its terms, and the rectangle splits into four smaller pieces. Add the pieces to get the product.
See the skill
(x + 3)(x + 2) as four rectangles
x² + 2x + 3x + 6 = x² + 5x + 6
Factoring runs the area model backward
To factor \(\displaystyle x^2 + 5x + 6\), find two numbers that multiply to 6 and add to 5: 2 and 3. So \(\displaystyle x^2 + 5x + 6 = (x + 2)(x + 3)\). Signs matter: for \(\displaystyle x^2 - 9x + 20\), you need two numbers that multiply to 20 and add to −9, which are −4 and −5.
Always look for a greatest common factor first. \(\displaystyle 6x^2 + 9x = 3x(2x + 3)\). Factoring out 3x first leaves a simpler expression in the parentheses.
Patterns worth recognizing on sight
A few patterns appear again and again. Recognizing them saves several steps.
See the skill
Special products
Difference of squares
a² − b² = (a + b)(a − b)
9x² − 16 = (3x + 4)(3x − 4)
Perfect square (sum)
(a + b)² = a² + 2ab + b²
(x + 5)² = x² + 10x + 25
Perfect square (difference)
(a − b)² = a² − 2ab + b²
(2x − 3)² = 4x² − 12x + 9
Common trap
(a + b)² ≠ a² + b²
Do not forget the middle term 2ab.
Your turn
Quick practice
Which expression is equivalent to 9x² − 16?
Hint: Both terms are perfect squares, and they are subtracted.
Check equivalence with a number
If two expressions are equivalent, they give the same value for every x. Substitute a simple number into the original and each choice. For \(\displaystyle (x + 3)(x + 2)\) at \(\displaystyle x = 2\): \(\displaystyle 5 \times 4 = 20\), and \(\displaystyle x^2 + 5x + 6 = 4 + 10 + 6 = 20\). Any choice that does not give 20 is wrong.
Avoid 0 and 1 as test values, because they can make different expressions look equal. If two choices both match, test a second number.
Practice: try these before checking the answers
Expand or factor, then check one answer by substituting x = 2.
- 1. Expand \(\displaystyle (x - 4)(x + 7)\).
- 2. Factor \(\displaystyle x^2 - 9x + 20\).
- 3. Factor \(\displaystyle 4x^2 - 25\).
- 4. Factor \(\displaystyle 12x^3 - 8x^2\).
Sources & further reading
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