Let’s solve each problem step by step. We’ll use algebra to find the dimensions based on area and relationships between length/width or base/height.
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Problem 1: Rectangle with length 6 inches more than width, area = 91 in²
Let:
- Width = x
- Length = x + 6
Area of rectangle = length × width
So:
x(x + 6) = 91
→ x² + 6x = 91
→ x² + 6x - 91 = 0
Now factor this quadratic:
We need two numbers that multiply to -91 and add to 6 → 13 and -7
So: (x + 13)(x - 7) = 0
Solutions:
x = -13 (reject — can’t have negative width)
x = 7 → width = 7 in, length = 7 + 6 = 13 in
✔ Check: 7 × 13 = 91 → correct!
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Problem 2: Rectangular garden, length is 4 meters more than width, area = 60 m²
Let:
- Width = x
- Length = x + 4
Area = x(x + 4) = 60
→ x² + 4x = 60
→ x² + 4x - 60 = 0
Factor: Find two numbers that multiply to -60 and add to 4 → 10 and -6
So: (x + 10)(x - 6) = 0
Solutions:
x = -10 (reject)
x = 6 → width = 6 m, length = 6 + 4 = 10 m
✔ Check: 6 × 10 = 60 → correct!
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Problem 3: Parallelogram, base is 7 cm more than altitude, area = 30 cm²
Let:
- Altitude (height) = x
- Base = x + 7
Area of parallelogram = base × height
So:
x(x + 7) = 30
→ x² + 7x = 30
→ x² + 7x - 30 = 0
Factor: Two numbers that multiply to -30 and add to 7 → 10 and -3
So: (x + 10)(x - 3) = 0
Solutions:
x = -10 (reject)
x = 3 → altitude = 3 cm, base = 3 + 7 = 10 cm
✔ Check: 3 × 10 = 30 → correct!
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Final Answer:
1) Width = 7 in, Length = 13 in
2) Width = 6 m, Length = 10 m
3) Altitude = 3 cm, Base = 10 cm
Parent Tip: Review the logic above to help your child master the concept of quadratic equation word problems worksheet with answers.
