MATHEMATICS

Mathematics



Mathematics

1.1

(a)

Factorize the left side:

x(x+5) = 0

This is true if either x = 0 or (x+5) = 0

if x = 0 then the value of x = 0 (simple)

if (x + 5) = 0 then x = -5

So the roots/solution of the equation

is x = 0, -5

(b)

-20x= -8x^2

(add the -8x^2 to the side with the -20x)

8x^2 - 20x=0

(now factor what does 8x^2 and -20x have in common?)

4x(2x-5)=0

( now you want to make each individual part equal to zero)

4x=0 (divide by four) x=0

(now do the parentheses) (2x-5)=0 (add the five to the right side) so, 2x=5 ( now divide by 2) x=5/2

so, x=0 and x=5/2

(c)

For a polynomial of the form x^(2)+bx+c, find two factors of c (-15) that add up to b (-2). In this problem 3*-5=-15 and 3-5=-2, so insert 3 as the right hand term of one factor and -5 as the right-hand term of the other factor.

(x+3)(x-5)=0

Set each of the factors of the left-hand side of the equation equal to 0.

x+3=0_x-5=0

Since 3 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 3 from both sides.

x=-3_x-5=0

Set each of the factors of the left-hand side of the equation equal to 0.

x=-3_x-5=0

Since -5 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 5 to both sides.

x=-3_x=5

The complete solution is the set of the individual solutions.

x=-3,5

1.2

(a)

3x(2x-5) +1(2x-5)

(2x-5) (3x+1)

x=5/2 or x= -1/3

(b)

1. x-2 / 3x-12 = 0

x-2=0

x=2

2. 5x / x(x+5) = 0

x cancels

5 doesn't = 0

3. x-6 / (x-3)(x+8) = 0

x-6=0

x=6

(c)

x^2 - 3*x - 18 = 0 ,divide both side with 1

So we get x^2 - 3*x - 18 = 0 ,

Which means that the coefficient of x is -3

We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = -3/2 = -1.5

Which means we can turn the equation into x^2 - 3*x + 2.25 - 20.25 = 0

And it is the same with ( x - 1.5 )^2 - 20.25 = 0

Which is the same with (( x - 1.5 ) - 4.5 ) * (( x - 1.5 ) + 4.5 ) = 0

And it is the same with ( x - 1.5 - 4.5 ) * ( x - 1.5 + 4.5 ) = 0

Do the addition/subtraction, and we get ( x - 6 ) * ( x + 3 ) = 0

The answers are x1 = 6 and x2 = -3

1.3

(a)

3x + 5x - 10 = 4 - 2x - 1

8x - 10 = 3 - 2x

8x + 2x = 3 + 10

10x = 13

x = 13/10

(b)

let y= x^2 then think of it as:

y^2-3y-4 then you have:

(y+1)(y-4)

so:

(x^2 +1 ) (x^2 -4)

now keep factoring:

(x^2 +1)(x-2)(x+2)

For high school math you'd stop here but, you can factor it more if you are using imagery numbers...

(x-i)(x+i)(x-2)(x+2)

where i = sqrt(-1)

1.4

(a)

Half the x co-efficient, and modify the constant accordingly:

(x-4)^2 = x^2-8x+16

We need to -26 to get to the -10 seen, ...