Solve (a) (r - 9u)² (b) 4(a + b)(a - 2b)(b - a) Simplify (a) -12 (m+9) = 396 (b) 6u - 18 = 8u/40 - 3½ (c) 8c - 100 = -14c + 308

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

a) (r-9u)^2=r^2 - 2*9u*r + (9u)^2=

=(r^2)-18*(u*r) + 81*(u^2)

I have to metion that it was used the formula

(a-b)^2=a^2 - 2*a*b - b^2

 

b) For solving this relation, we have to re-write it, based on the property of commutativity of adding relation, so that (a+b) could be written as (b+a). The relation will become:

4(b+a)*(a-2b)*(b-a)

In this way, we could note that we have the development of squares difference: (b+a)*(b-a)=(b)^2-(a)^2

4(b+a)*(a-2b)*(b-a)=4[(b)^2-(a)^2]*(a-2b)=

We'll open the parenthesis and we'll have:

4[(b)^2-(a)^2]*(a-2b)=4a*b^2-8b^3-4a^3+8a^2*b

 

SIMPLIFY

a)-12(m+9)=396

We'll divide with (-12), the relation above:

(m+9)=(396/-12)

m+9=-33

m=-33-9

m=-42

b)6u-18=8u/40- 3(1/2)

First of all, we note that "8u/40" could be written as "8u/8*5", in order to simplify 8. The result will be "u/5".

Second of all, we'll write the terms with "u" in the left side of equal and the free terms on the right side:

6u-(u/5)=18-(7/2)

We'll find out the same denominator on the left side, which is 5, and we'll amplify "6u" with 5.

(5*6u-u)/5=18-(7/2)

We'll find out the same denominator on the right side, which is 2, and we'll amplify "18" with 2.

(30u-u)/5=(36-7)/2

29u/5=29/2

We could divide the relation with "29"

u/5=1/2

We'll cross multiplying:

2u=5

u=5/2

u=2.5

c)8c-100=-14c+308

As we've did at the point b), we'll write the terms with "u" in the left side of equal and the free terms on the right side:

8c+14c=308+100

22c=408

We could simplify with 2, because both 22 and 204 are divisible by 2:

11c=204

c=204/11

c=18.(54)

 

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

a)

We use (a+b)^2=a^2+2ab+b^2 and expand the expression(r-9u)^2

a=r and b=-9u. Therefore(r-9u)^2=a^2-2r*(-u)+(-9u)^2= a^2-18au+81u^2.

b)

4(a+b)(a-2b)((b-a) could be written like:-4(a+b)(a-2b)((a-b)

We Use, the identity ,(x+l)(x+m)(x+n)=x^3+(l+m+n)x^2+(lm+mn+cnl)x+lmn, we get  by putting x=a, l=b, m=-2b and n=-b, we get:

-4{a^3+(b-2b-b)a^2+[(b)(-2b)+(-2b)(-b)+(-b)(b)]a+(b)(-2b)(-b)}

=-4{a^3-2ba^2-b^2a+2b^3}

=-4a^3+8a^2b+4ab^2-8b^3

Simplifying and solving for the unkown:

a)

-12 (m+9) = 396

Divide both sides by -12  and then subtract -9 from both sides to get m.

m+9 = 396/(-12)=-33

m= -33-9= -42

m=-42

b)

6u - 18 = 8u/40 - 3½

8u/40=0.2u and   3  1/2 = 3.5

We use the above ordinary decimal numbers,  instead of fractions and rewrite the equation:

6u-18= 0.2u -3.5.

Collect the  variables on left and numerals on the right by adding , -0.2u+18 to both sides:

6u-18-0.2u+18=0.2u-3.5-0.2u+18

Simplify both sides:

6u-0.2u= -3.5+18

Simplify bith sides:

5.8u=14.

Divide both sides by 5.8:

u=14.5/5.8=2.5 or

u=2.5

c)

8c - 100 = -14c + 308

Collect  the variables on left and numbers, the known on the right, by adding 14c+100 to both sides of the equation and simplify:

8c-100+14c-100=-14c+308+14c+100

Simplify:

22c=408

Divide by 22 both sides:

c=408/22=18.545454...

c=18.545454..

 

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