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Quizzes > High School Quizzes > Mathematics

Factoring Difference of Squares Practice Quiz

Practice factoring quadratic expressions with guided answers

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art representing a high school algebra quiz about factoring differences of squares.

Factor the expression: x^2 - 9
(x+3)(x-3)
(x+1)(x-9)
(x-3)^2
(x+9)(x-1)
The expression x^2 - 9 is a difference of squares because 9 is 3^2. Applying the formula a^2 - b^2 = (a+b)(a-b) with a = x and b = 3 yields (x+3)(x-3).
Factor the expression: 4x^2 - 25
(2x+5)(2x-5)
4x^2 - 25
(2x+5)^2
(2x-5)^2
Here, 4x^2 is the square of 2x and 25 is the square of 5. Using the difference of squares rule, the expression factors into (2x+5)(2x-5).
Factor the expression: 9y^2 - 16
(3y+4)^2
(3y-4)^2
(3y+4)(3y-4)
(9y+16)(y-1)
Since 9y^2 is (3y)^2 and 16 is 4^2, the expression is a difference of squares. It factors neatly into (3y+4)(3y-4) using the formula a^2 - b^2 = (a+b)(a-b).
Factor the expression: x^2 - 49
(x+7)(x-7)
(x+1)(x-49)
(x+49)(x-1)
(x-7)^2
Here, 49 is the square of 7. Recognizing x^2 as x^2 and 49 as 7^2, the expression factors using the difference of squares to (x+7)(x-7).
Factor the expression: 25z^2 - 64
(25z+64)(z-1)
(5z+8)(5z-8)
(5z+64)(z-8)
(5z-8)^2
The term 25z^2 is (5z)^2 and 64 is 8^2. Using the difference of squares formula, we factor the expression as (5z+8)(5z-8).
Factor the expression: 16x^2 - 9
(4x-3)^2
(16x+9)(x-1)
(4x-9)(4x+3)
(4x+3)(4x-3)
Recognize that 16x^2 is (4x)^2 and 9 is 3^2. Applying the difference of squares formula a^2 - b^2 = (a+b)(a-b) produces (4x+3)(4x-3).
Factor the expression: 49a^2 - 64b^2
(7a+8b)^2
(7a+8b)(7a-8b)
(7a-8b)^2
49a^2 - 64b^2
Both terms are perfect squares: 49a^2 equals (7a)^2 and 64b^2 equals (8b)^2. The difference of squares formula then gives the factorization (7a+8b)(7a-8b).
Factor the expression: 36m^2 - 49n^2
(6m+49n)(6m-7n)
(6m+7n)(6m-7n)
(6m-7n)^2
(36m+49n)(m-n)
Since 36m^2 is (6m)^2 and 49n^2 is (7n)^2, the expression is a difference of squares. Therefore, factoring it gives (6m+7n)(6m-7n).
Factor the expression: 9x^2 - 100
(3x-10)^2
(3x+10)^2
(9x+100)(x-1)
(3x+10)(3x-10)
Here, 9x^2 can be written as (3x)^2 and 100 as 10^2. The difference of squares formula then factors the expression to (3x+10)(3x-10).
Factor the expression: 4y^2 - 49
(2y+7)(2y-7)
(2y+7)^2
(2y-7)^2
(4y+49)(y-1)
Since 4y^2 is (2y)^2 and 49 is 7^2, applying the difference of squares formula yields (2y+7)(2y-7).
Factor completely: x^4 - 16
(x^2+4)(x+2)(x-2)
x^4 - 16
(x^2-4)(x^2+4)
(x+4)(x-4)(x+2)(x-2)
First, view x^4 - 16 as (x^2)^2 - 4^2, which factors into (x^2+4)(x^2-4). Then, note that x^2-4 is itself a difference of squares and factors as (x+2)(x-2), giving the complete factorization.
Factor the expression: 49 - 9a^2
49 - 9a^2
(7+3a)(7-3a)
(7-3a)^2
(7+3a)(3a-7)
Recognize that 49 is 7^2 and 9a^2 is (3a)^2; hence the expression is a difference of squares. This factors to (7+3a)(7-3a) using the standard formula.
Factor the expression: 64x^2 - 81
(8x+9)(8x-9)
(8x-9)^2
(8x+81)(x-9)
64x^2 - 81
Since 64x^2 is (8x)^2 and 81 is 9^2, applying the difference of squares rule factors the expression as (8x+9)(8x-9).
Factor the expression: 4x^4 - 9y^4
(4x^2+9y^2)(x^2-y^2)
(2x^2+3y^2)(2x^2-3y^2)
(2x+3y)(2x-3y)
4x^4 - 9y^4
Write 4x^4 as (2x^2)^2 and 9y^4 as (3y^2)^2 to reveal the difference of squares structure. The expression then factors into (2x^2+3y^2)(2x^2-3y^2).
Factor the expression: 81 - 16z^2
(9-4z)^2
(4z+9)(16z-9)
(9+4z)(9-4z)
81 - 16z^2
Since 81 is 9^2 and 16z^2 is (4z)^2, the difference of squares formula gives the factorization (9+4z)(9-4z).
Factor completely: 16x^4 - 81y^4
16x^4 - 81y^4
(4x^2+9y^2)(2x+3y)(2x-3y)
(4x^2+9y^2)(4x^2-9y^4)
(2x+3y)(2x-3y)(4x^2-9y^2)
Begin by recognizing 16x^4 as (4x^2)^2 and 81y^4 as (9y^2)^2, which makes the expression a difference of squares: (4x^2+9y^2)(4x^2-9y^2). Then, factor the second term further as (2x+3y)(2x-3y) to achieve complete factorization.
Factor completely: x^4 - 81
(x+3)(x-3)
(x^2+9)(x+3)(x-3)
(x+9)(x-9)(x+3)(x-3)
(x^2-9)(x^2+9)
Express x^4 - 81 as (x^2)^2 - 9^2 to apply the difference of squares formula, which gives (x^2+9)(x^2-9). Then, notice that x^2-9 is itself a difference of squares factoring to (x+3)(x-3), resulting in the complete factorization.
Factor completely: 25x^4 - 100x^2
25x^4 - 100x^2
(5x+2)(5x-2)(x+2)(x-2)
25x^2(x+2)(x-2)
25x^2(x^2-4)
Start by factoring out the greatest common factor, 25x^2, which leaves 25x^2(x^2-4). Then, recognize that x^2-4 is a difference of squares and factors into (x+2)(x-2), giving the complete factorization.
Factor completely: 4x^4 - 9y^2
(4x^2+9y)(x^2-y)
4x^4 - 9y^2
(2x+3y)(2x-3y)
(2x^2+3y)(2x^2-3y)
Notice that 4x^4 can be written as (2x^2)^2 and 9y^2 as (3y)^2. Applying the difference of squares formula then factors the expression to (2x^2+3y)(2x^2-3y).
Factor completely: 16x^4 - 81
(4x^2-9)(2x+3)(2x-3)
(4x^2+9)(2x+3)(2x-3)
16x^4 - 81
(4x^2+9)(4x^2-9)
First, rewrite 16x^4 - 81 as (4x^2)^2 - 9^2, which factors to (4x^2+9)(4x^2-9). Then, notice that 4x^2-9 is a difference of squares and factors further into (2x+3)(2x-3), yielding the complete factorization.
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Study Outcomes

  1. Understand the structure of a difference of squares in algebraic expressions.
  2. Identify terms that can be rewritten as perfect squares in algebraic expressions.
  3. Apply the difference of squares factoring technique to simplify expressions.
  4. Analyze quadratic expressions to determine if the difference of squares method is applicable.
  5. Verify the accuracy of factored expressions by expanding them back to the original form.

Factoring Quiz: Squares & Quadratics Cheat Sheet

  1. Master the formula - The heart of it all is a² − b² = (a+b)(a−b). Visualize it as two friends splitting up in pairs! Once you get it, factoring becomes a breeze. Difference of Two Squares (Wikipedia)
  2. Spot those perfect squares - Anytime you see x², 4y², or 9, think "square power." Recognizing them fast is like having a secret algebra spy in your toolbox. SchoolTube Step-by-Step Guide
  3. Check for subtraction only - The magic works only when you subtract squares, not add them. Remember, x² − 16 fits, but x² + 9 defies this trick! MathDoubts Tutorial
  4. Crunch simple examples first - Start with easy ones like x² − 9 = (x+3)(x−3). Getting comfy here builds your factoring muscles for tougher cases. ChiliMath Practice Problems
  5. Level up with complex expressions - Try 4x² − 9y²; it factors into (2x+3y)(2x−3y). Think of it as a bigger version of the same neat trick! OnlineMathLearning Advanced Examples
  6. Simplify algebraic puzzles - Factoring differences makes solving equations sleek. When you see messy expressions, break them into squares to tackle them step by step. MathDoubts Worksheet
  7. Mind the sum of squares trap - You can't factor a² + b² using this method. That's a whole different beast - reserve this formula for subtraction only! More on Difference vs. Sum (Wikipedia)
  8. Verify by expanding back - Always expand your factors to ensure they recombine into the original expression. It's your safety net against sneaky mistakes! SchoolTube Verification Tips
  9. Practice across various problems - Jump into mixed exercises to challenge your brains. Variety builds confidence and cements your new factoring skills! More ChiliMath Drills
  10. Tap into extra resources - From worksheets to interactive quizzes, keep exploring different platforms. Extra practice is the secret sauce to mastery! Twinkl Practice Worksheet
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