H Algebra 2/Trig · Unit 0
Shaded area is the big rectangle minus any cut-out squares. Then factor if you can.
Shaded area
Area of a rectangle = length × width. If squares are cut out, subtract those areas. Then factor: pull out a GCF, and use difference of squares when you see a² − b².
Worked example
An 18 by 8 rectangle with four x-by-x corners cut out:
18 · 8 = 144. Four squares: 4x².
144 − 4x² = 4(36 − x²) = 4(6 − x)(6 + x)
The blue rectangle is (x + 3) by 5. Write the area in factored form.
Area of a rectangle is length times width.
Area = 5(x + 3).
Answer: 5(x + 3)
The rectangle is (x + 4) by 6. Write the area in factored form.
Area = 6(x + 4).
Answer: 6(x + 4)
The rectangle is (x + 2) by 7. Write the area in factored form.
Area = 7(x + 2).
Answer: 7(x + 2)
A 9 by 5 rectangle has one x-by-x square cut out. Write the shaded area.
Big rectangle: 9 · 5 = 45.
Subtract the square: 45 − x².
Answer: 45 − x²
An 8 by 4 rectangle has one x-by-x square cut out. Write the shaded area.
8 · 4 = 32.
Subtract x².
Answer: 32 − x²
The rectangle is (x + 6) by 3. Write the area in factored form.
Area = 3(x + 6).
Answer: 3(x + 6)
A 12 by 10 rectangle has two x-by-x squares cut out. Write the shaded area and factor out a GCF.
Big rectangle: 12 · 10 = 120.
Two squares: 2x².
Shaded: 120 − 2x² = 2(60 − x²).
Answer: 2(60 − x²)
A 15 by 8 rectangle has two x-by-x squares cut out. Write the shaded area and factor out a GCF.
15 · 8 = 120. Subtract 2x².
120 − 2x² = 2(60 − x²).
Answer: 2(60 − x²)
A 16 by 6 rectangle has two x-by-x squares cut out. Write the shaded area and factor out a GCF.
16 · 6 = 96. Subtract 2x².
96 − 2x² = 2(48 − x²).
Answer: 2(48 − x²)
A 14 by 6 rectangle has two x-by-x squares cut out. Write the shaded area and factor out a GCF.
14 · 6 = 84. Subtract 2x².
84 − 2x² = 2(42 − x²).
Answer: 2(42 − x²)
An 11 by 9 rectangle has one x-by-x square cut out. Write the shaded area.
11 · 9 = 99.
Shaded: 99 − x².
Answer: 99 − x²
A 13 by 4 rectangle has one x-by-x square cut out. Write the shaded area.
13 · 4 = 52.
Shaded: 52 − x².
Answer: 52 − x²
Write an expression in factored form for the area of the shaded region. The four corners are x-by-x squares.
Big rectangle: 18 · 8 = 144.
Four squares: 4x².
Shaded: 144 − 4x².
Factor 4: 4(36 − x²).
36 − x² is a difference of squares: (6 − x)(6 + x).
4(6 − x)(6 + x)
Answer: 4(6 − x)(6 + x)
Four x-by-x corners are cut from a 16 by 4 rectangle. Write the shaded area in factored form.
16 · 4 = 64. Subtract 4x².
64 − 4x² = 4(16 − x²) = 4(4 − x)(4 + x).
Answer: 4(4 − x)(4 + x)
Four x-by-x corners are cut from a 10 by 10 square. Write the shaded area in factored form.
10 · 10 = 100. Subtract 4x².
100 − 4x² = 4(25 − x²) = 4(5 − x)(5 + x).
Answer: 4(5 − x)(5 + x)
Four x-by-x corners are cut from a 12 by 3 rectangle. Write the shaded area in factored form.
12 · 3 = 36. Subtract 4x².
36 − 4x² = 4(9 − x²) = 4(3 − x)(3 + x).
Answer: 4(3 − x)(3 + x)
One x-by-x square is cut from an 11 by 11 square. Write the shaded area in factored form.
11 · 11 = 121. Subtract x².
121 − x² = (11 − x)(11 + x).
Answer: (11 − x)(11 + x)
Four x-by-x corners are cut from an 8 by 8 square. Write the shaded area in factored form.
8 · 8 = 64. Subtract 4x².
64 − 4x² = 4(16 − x²) = 4(4 − x)(4 + x).
Answer: 4(4 − x)(4 + x)
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