Quadratic Factorisation
The quadratic form is ax² + bx + c. In the equation x² - 8x + 12, c = 12. Are the factors of c = (+6) × (+2), c = (-6) × (-2), or c = (+2) × (-6)?
In x² - 8x + 12, because b = -8 (a negative number), c should be factored in (-2) and (-6)
The quadratic form is ax² + bx + c. In the equation x² + 3x - 4, c = -4. Are the factors of c = (+2) × (+2), c = (-2) × (-2), or c = (4) × (-1)?
In x² + 3x - 4, because b does not matter as much as the sign of c, c should be factored into opposite signs (+4) and (-1)
In the equation -x² + 7x - 10, what should be factored out first?
Factor out (-1) to simplify calculations: -x² + 7x - 10 = (-1)(x² - 7x + 10)
In the equation -2x² + 5x + 12, what should be factored out first?
Factor out (-1) to simplify calculations: -2x² + 5x + 12 = (-1)(2x² - 5x - 12)
In the equation -2x² + x + 10, what should be factored out first?
Factor out (-1) to turn a negative coefficient of -x² into a positive coefficient of x²: -2x² + x + 10 = (-1)(2x² - x - 10)
To simplify calculations, what should be factored out first in 2x² + 4x - 6?
Factor out (+2) to simplify calculations: 2x² + 4x - 6 = (2)(x² + 2x - 3)
To simplify calculations, what should be factored out first in 3x² - 9x - 30?
Factor out (+3) to simplify calculations: 3x² - 9x - 30 = (3)(x² - 3x - 10)
Can 7x² - 20xy - 3y be simplified before normal factorisation?
Not really. Use a grid to factor 7x² - 20xy - 3y² into (7x + y)(x - 3y)
Can 5(m²)(n²) - 31mn + 6 be simplified before normal factorisation?
Let mn be a. Rewrite as 5a² - 31a + 6 before you factor into... (5mn - 1)(mn - 6)
Can 2a² - 5ab - 3b² be simplified before normal factorisation?
Not really. Use a grid to factor 2a² - 5ab - 3b² into (2a + b)(a - 3b). Note that special identities cannot be used because there is no 2ab term
What is the difference of squares identity?
a² - b² = (a + b)(a - b) is called the difference of squares
What is the square of a sum identity?
(a + b)² = a² + 2ab + b² is called a square of a sum
What is the square of a difference identity?
(a - b)² = a² - 2ab + b² is called a square of a difference
a² - 2ab + b² = ??
It is the Algebra Identity of a Square of a difference: a² - 2ab + b² = (a - b)²
a² + 2ab + b² = ??
It is the Algebra Identity of a Square of a sum: a² + 2ab + b² = (a + b)²
(a + b)(a - b) = which identity?
(a + b)(a - b) = difference of squares identity = a² - b²
If (x + y)² = x² + y² + 2xy, then what does x² + y² = ??
The identity of this equation (x + y)² = x² + y² + 2xy is a square of a sum, and it can be rewritten as x² + y² = (x + y)² - 2xy
If (x - y)² = x² + y² - 2xy, then what does x² + y² = ??
The identity of this equation (x - y)² = x² + y² - 2xy is a square of a difference, and it can be rewritten as x² + y² = (x - y)² + 2xy
Square A has a side length of x, while Square B has a side length of y. From the algebraic identities, how many ways can their areas be added together?
The square of a sum (x + y)² = x² + y² + 2xy can be rewritten as x² + y² = (x + y)² - 2xy. The square of a difference (x - y)² = x² + y² - 2xy can be rewritten as x² + y² = (x - y)² + 2xy
Factor x² + 10x + 25. Can you notice a special identity?
a² + 2ab + b² = (a + b)² can be identified in x² + 10x + 25. Thus, x² + 10x + 25 = (x + 5)²
Factor 4 + 12xy + 9x²y². Can you notice a special identity?
a² + 2ab + b² = (a + b)² can be identified in 4 + 12xy + 9(x²)(y²). Thus 9(x²)(y²) + 12xy + 4 = (2 + 3xy)². Notice that 9(x²)(y²) = 3xy
Factor 49t² - 14t + 1. Is there a special identity?
a² - 2ab + b² = (a - b)² can be identified in 4t² - 14t + 1 to factor into (7t - 1)²
Factor 4x² - 36xy + 81y². Is there a special identity?
a² - 2ab + b² = (a - b)² can be identified for factorisation. Notice that 4x² = (2x)² and 81y² = (9y)² in 4x² - 36xy + 81y². Thus, 4x² - 36xy + 81y² = (2x - 9y)²
Factor z² - 16. Which special identity can be used?
Difference of squares: a² - b² = (a + b)(a - b). z² - 16 = (z + 4)(z - 4)
Factor 49m² - 9n². Which special identity can be used?
Difference of squares: a² - b² = (a + b)(a - b). 49m² - 9n² = (7m + 3n)(7m - 3n)
Factor 3x² - 75. Can a special identity be used? Should the equation be simplified first?
Factor out a 3 first: 3x² - 75 = (3)(x² - 25). The difference of squares can now be used: 3x² - 75 = (3)(x + 5)(x - 5)
Factor 5t² - 20. Can a special identity be used? Should the equation be simplified first?
Factor out a 5 first: 5t² - 20 = 5(t² - 4), then factor with the difference of squares: 5t² - 20 = 5(t + 2)(t - 2)
Evaluate 238² - 237² without a calculator.
Difference of squares: (a + b)² = (a + b)(a - b). 238² - 237² = (238 - 237)(238 + 237) = 475 x 1 = 475
What is factoring by grouping?
Grouping terms with common factors and factoring out the GCF of each group
What is factoring trinomials of the form ax² + bx + c?
A tri-nomial of ax² + bx + c is factored as such: Find two numbers that multiply to ac and add to b, then factoring as (mx + n)(px + q)
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