Examples with Mathspeak

Compute Area of House

First find the area of the square in square feet. A = s 2 = ( 2 x ) 2 = 4 x 2 A = s 2 = (2x) 2 = 4 x 2

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Then find the area of the triangle in square feet. A = 1 2 b h = 1 2 ( 2 x ) ( 3 2 ) = 3 2 x A = 1 2 bh = 1 2 (2x)( 3 2 ) = 3 2 x
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Next find the area of the rectangular door in square feet. A = l w = x ⋅ 1 = x A = lw = x⋅1 = x
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Perfect Square Trinomials

  ( x + 5 ) 2 = x 2 + 10 x + 25 ( x − 3 ) 2 = x 2 − 6 x + 9 ( 4 x − 1 ) 2 = 16 x 2 − 8 x + 1   (x+5) 2 = x 2 +10x+25 (x−3) 2 =    x 2 −6x+9 (4x−1) 2 = 16 x 2 −8x+1

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Difference of Squares

( x + 1 ) ( x − 1 ) = x 2 − x + x − 1 = x 2 − 1 (x+1)(x−1) = x 2 −x+x−1 = x 2 −1

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The middle term drops out, resulting in a difference of squares. ( x + 5 ) ( x − 5 ) = x 2 − 25 ( x + 11 ) ( x − 11 ) = x 2 − 121 ( 2 x + 3 ) ( 2 x − 3 ) = 4 x 2 − 9 (x+5)(x−5) = x 2 −25 (x+11)(x−11) = x 2 −121 (2x+3)(2x−3) = 4 x 2 −9
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Performing Operations with Polynomials of Several Variables

( a + 2 b ) ( 4 a − b − c ) a ( 4 a − b − c ) + 2 b ( 4 a − b − c ) Use the distributive property . 4 a 2 − a b − a c + 8 a b − 2 b 2 − 2 b c Multiply . 4 a 2 + ( − a b + 8 a b ) − a c − 2 b 2 − 2 b c Combine like terms . 4 a 2 + 7 a b − a c − 2 b c − 2 b 2 Simplify . (a+2b)(4a−b−c) a(4a−b−c)+2b(4a−b−c) Use the distributive property. 4 a 2 −ab−ac+8ab−2 b 2 −2bc Multiply. 4 a 2 +(−ab+8ab)−ac−2 b 2 −2bc Combine like terms. 4 a 2 +7ab−ac−2bc−2 b 2 Simplify.

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