Evaluate 2(x+3y)(x^2-3xy+9y^2)+2(x-3y)(x^2+3xy+9y^2)/x^2 if x equals 2 3/4 and y=-1. a. -1 b. 2...

Question:

Evaluate {eq}\frac{2(x+3y)(x^2-3xy+9y^2)+2(x-3y)(x^2+3xy+9y^2)}{x^2} {/eq} if x=2{eq}\frac{3}{4} {/eq} and y=-1.

a. -1

b. 2{eq}\frac{3}{4} {/eq}

c. 5{eq}\frac{1}{2} {/eq}

d. 8{eq}\frac{1}{4} {/eq}

e. 11

Simplifying criteria:

To simplify any mathematical expression is converting from one equal form to another such that second form is less complex to the first one.

Following one rule which we used in the given example:

Multiply Fractions:

{eq}x\frac{y}{z} = \frac{{xy}}{z} {/eq}

Answer and Explanation:

Given that: {eq}\displaystyle \frac{{2(x + 3y)({x^2} - 3xy + 9{y^2}) + 2(x - 3y)({x^2} + 3xy + 9{y^2})}}{{{x^2}}} {/eq}

{eq}\displaystyle \eqalign{ & \frac{{2(x + 3y)({x^2} - 3xy + 9{y^2}) + 2(x - 3y)({x^2} + 3xy + 9{y^2})}}{{{x^2}}} \cr & {\text{Let,}} \cr & f(x,y) = \frac{{2(x + 3y)({x^2} - 3xy + 9{y^2}) + 2(x - 3y)({x^2} + 3xy + 9{y^2})}}{{{x^2}}} \cr & f(x,y){\text{ at }}x = \frac{{11}}{4},y = - 1; \cr & f(\frac{{11}}{4}, - 1) = \frac{{2\left( {\left( {\frac{{11}}{4}} \right) + 3\left( { - 1} \right)} \right)\left( {{{\left( {\frac{{11}}{4}} \right)}^2} - 3\left( {\frac{{11}}{4}} \right)\left( { - 1} \right) + 9{{\left( { - 1} \right)}^2}} \right) + 2\left( {\left( {\frac{{11}}{4}} \right) - 3\left( { - 1} \right)} \right)\left( {{{\left( {\frac{{11}}{4}} \right)}^2} + 3\left( {\frac{{11}}{4}} \right)\left( { - 1} \right) + 9{{\left( { - 1} \right)}^2}} \right)}}{{{{\left( {\frac{{11}}{4}} \right)}^2}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2\left( {\frac{{11}}{4} - 3} \right)\left( {\left( {\frac{{121}}{{16}}} \right) + \frac{{33}}{4} + 9} \right) + 2\left( {\frac{{11}}{4} + 3} \right)\left( {\frac{{121}}{{16}} - \frac{{33}}{4} + 9} \right)}}{{\frac{{121}}{{16}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2\left( {\frac{{11 - 12}}{4}} \right)\left( {\frac{{121}}{{16}} + \frac{{33}}{4} + 9} \right) + 2\left( {\frac{{11 + 12}}{4}} \right)\left( {\frac{{121}}{{16}} - \frac{{33}}{4} + 9} \right)}}{{\frac{{121}}{{16}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2\left( {\frac{{ - 1}}{4}} \right)\left( {\frac{{121 + 132 + 144}}{{16}}} \right) + 2\left( {\frac{{23}}{4}} \right)\left( {\frac{{121 - 132 + 144}}{{16}}} \right)}}{{\frac{{121}}{{16}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\left( {\frac{{ - 1}}{2}} \right)\left( {\frac{{397}}{{16}}} \right) + \left( {\frac{{23}}{2}} \right)\left( {\frac{{133}}{{16}}} \right)}}{{\frac{{121}}{{16}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\left( {\frac{{ - 397 + 3059}}{{32}}} \right)}}{{\frac{{121}}{{16}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\left( {\frac{{2662}}{2}} \right)}}{{\frac{{121}}{1}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1331}}{{121}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 11 \cr & \cr & f(\frac{{11}}{4}, - 1) = 11 \cr & {\text{Option }}\left( e \right){\text{ is correct}}{\text{.}} \cr} {/eq}


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How to Simplify an Addition Expression

from 6th-8th Grade Math: Practice & Review

Chapter 8 / Lesson 1
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