SOLUTION: Divide a Rational Expression
Can you please show it in steps
2x2 + 5x – 12 ÷ 2x2 – 7x + 6
9x2 – 16 3x2 – x – 4
Algebra ->
Rational-functions
-> SOLUTION: Divide a Rational Expression
Can you please show it in steps
2x2 + 5x – 12 ÷ 2x2 – 7x + 6
9x2 – 16 3x2 – x – 4
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You can put this solution on YOUR website! 1) 2x2 + 5x – 12 ÷ 2x2 – 7x + 6
=(2x^2 + 5x – 12 )/( 2x^2 – 7x + 6)
= [(x+4)(2x-3)]/[(x-2)(2x-3)]
=(x+4)/(x-2)[(canceling (2x-3)]
2)(9x2 – 16 )/(3x2 – x – 4)
=[(3x)^2-(4)^2]/[(3x-4)(x+1)]
= [(3x+4)(3x-4)]/[(3x-4)(x+1)]
=(3x+4)/(x+1)[canceling (3x-4)]
Note: what was the main feature that was in play in these two problem?
Factorisable quadratic expressions!
Here are the factorisations.
2x^2 + 5x – 12
=2x^2 + (8x – 3x)-12
( writing the mid term as the sum of two terms whose product is the
product of the square term and the constant term][Here the product of the square term and the constant term is (2x^2)X(-12) = -24x^2
= -(1X2X2X2X3)x^2 =[ (2X2X2)x]X[-(3x)] = (8x)X(-3x)so that 5x= (8x-3x)
[the product is negative and the middle term is positive and hence the larger part positive and smaller part negative so that larger positive added to smaller negative is positive and (positive X negative) = negative]
=(2x^2 + 8x) – 3x-12 (by additive associativity)
=2x(x+4)-3(x+4)
=2xp-3p where p= (x+4)
=p(2x-3)
=(x+4)(2x-3)
( 2x^2 – 7x + 6)
=2x^2 – 4x-3x + 6 [the middle term (-7x) = (-4x)+(-3x)
such that (-4x)X(-3x) = 12x^2 = (2x^2)X(6)]
[the product is positive and the middle term is negative and hence both parts negative so that negative added to negative is negative and
(negative X negative) = positive]
=2x(x-2)-3(x-2)
=2xp-3p Where p=(x-2)
=p(2x-3)
=(x-2)(2x-3)
3x2 – x – 4
=3x2 – 4x+3x – 4 [ the middle term (-x) = (-4x)+(3x)
such that (-4x)X(3x) = -12x^2 = (3x^2)X(-4)]
[the product is negative and the middle term is negative and hence the larger part negative and smaller part positive so that larger negative added to smaller positive is negative and (negative X positive) = negative]
=x(3x-4)+1(3x-4)
=xp+p where p = (3x-4)
=p(x+1)
=(3x-4)(x+1)
(9x2 – 16 )
=[(3x)^2-(4)^2] (putting in the form [(a)^2-(b)^2] )
= [(3x+4)(3x-4)]
(using formula [(a)^2-(b)^2] = (a+b)(a-b) where here a =3x and b = 4)
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