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Questions and Answers
What indicates that both factors of c must be positive when factoring x² + bx + c?
What indicates that both factors of c must be positive when factoring x² + bx + c?
- b is negative and c is positive
- b is positive and c is positive (correct)
- b is zero and c is positive
- b is negative and c is negative
What would the values of p and q be when factoring x² + 4x - 21?
What would the values of p and q be when factoring x² + 4x - 21?
- p = -3, q = 7
- p = -4, q = 21
- p = 3, q = -7 (correct)
- p = 3, q = 7
What is the area of the square plot of land if its width is determined to be 60 meters?
What is the area of the square plot of land if its width is determined to be 60 meters?
- 3600 square meters (correct)
- 3000 square meters
- 2400 square meters
- 4800 square meters
Which expression represents the area of the pumpkin patch if its length is (s - 30) and width is (s - 40)?
Which expression represents the area of the pumpkin patch if its length is (s - 30) and width is (s - 40)?
What can be concluded when factoring a quadratic of the form x² + bx + c with c < 0?
What can be concluded when factoring a quadratic of the form x² + bx + c with c < 0?
Flashcards
Factoring x² + bx + c with positive 'c'
Factoring x² + bx + c with positive 'c'
When factoring a quadratic expression in the form x² + bx + c, if the constant term 'c' is positive, the factors 'p' and 'q' must have the same sign. This ensures that their product 'pq' is positive.
Factoring x² + bx + c with positive 'c' and 'b'
Factoring x² + bx + c with positive 'c' and 'b'
When factoring a quadratic expression in the form x² + bx + c, if the constant term 'c' is positive, the factors 'p' and 'q' must both be positive if the coefficient of the linear term 'b' is also positive. This ensures that their sum 'p + q' is positive.
Factoring x² + bx + c with negative 'c'
Factoring x² + bx + c with negative 'c'
When factoring a quadratic expression in the form x² + bx + c, if the constant term 'c' is negative, the factors 'p' and 'q' must have opposite signs. This ensures that their product 'pq' is negative.
Contextualizing Solutions to Quadratic Equations
Contextualizing Solutions to Quadratic Equations
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Calculating Area of Squares and Rectangles
Calculating Area of Squares and Rectangles
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Study Notes
Factoring x² + bx + c
- Factoring quadratic expressions of the form x² + bx + c, where 'c' is positive (and 'b' and 'c' are integers):
- If 'c' is positive, 'p' and 'q' (factors of the quadratic) have the same sign as 'b'.
- x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.
- Example:
- x² + 6x + 5 = (x + 1)(x + 5)
- x² - 6x + 5 = (x - 1)(x - 5)
Factoring x² + bx + c, when c is negative
- If 'c' is negative, 'p' and 'q' have different signs.
- Example:
- x² + 4x - 21 = (x - 3)(x + 7)
Applications of Factoring
- Example Problem:
- Area of a rectangular pumpkin patch embedded within a larger square plot of land.
- The patch's length = (s-30), width = (s-40).
- Area of patch = 600 m².
- Equation: 600 = (s - 30) * (s - 40).
- Solving the equation, the side length of the square lot = 60 meters.
- Area of square plot = 3600 m².
- Note: the smaller solution (s = 10 in above example) was rejected as illogical based on the visual diagram.
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Description
Explore the techniques of factoring quadratic expressions of the form x² + bx + c. This quiz covers cases when 'c' is both positive and negative, illustrated with examples. Additionally, discover practical applications of factoring in real-world problems.
