11. Factoring and solving equations
- A. Factor-
1. Factor 3x2 + 6x if possible.
Look for monomial (single-term) factors first; 3 is a factor of both 3x2
and 6x and so is x . Factor them out to get
3x2 + 6x = 3(x2 + 2x1 = 3x(x+ 2) .
2. Factor x2 + x - 6 if possible.
Here we have no common monomial factors. To get the x2 term
we'll have the form (x +-)(x +-) . Since
(x+A)(x+B) = x2 + (A+B)x + AB ,
we need two numbers A and B whose sum is 1 and whose product is
-6 . Integer possibilities that will give a product of -6 are
-6 and 1, 6 and -1, -3 and 2, 3 and -2.
The only pair whose sum is 1 is (3 and -2) , so the factorization is
x2 + x - 6 = (x+3)(x-2) .
3. Factor 4x2 - 3x - 10 if possible.
Because of the 4x2 term the factored form wli be either
(4x+A)(x +B) or (2x+A)(2x+B) . Because of the -10 the integer possibilities
for the pair A, B are
10 and -1 , -10 and 1 , 5 and -2 . -5 and 2 , plus each of
these in reversed order.
Check the various possibilities by trial and error. It may help to write
out the expansions
(4x + A)(x+ B) = 4x2 + (4B+A)x + A8
1 trying to get -3 here
(2x+A)(2x+B) = 4x2 + (2B+ 2A)x + AB
Trial and error gives the factorization 4x2 - 3x - 10 - (4x+5)(x- 2) .
4. Difference of two squares. Since (A + B)(A - B) = - B~ , any
expression of the form A' - B' can be factored. Note that A and B
might be anything at all.
Examples: 9x2 - 16 = (3x1' - 4' = (3x +4)(3x - 4)
x2 - 29 = x2 - (my)* = (x+ JTy)(x- my)
Range ⇒ Values that can 'come out' of the function.
As x → -∞, y → 0
As x → +∞, y → ∞
This means that the smallest value that can come out of the function is a value which is tiny but greater than 0 and the largest is ∞ so the range is option 1, all real numbers greater than 0.
(When I get these questions it always helps me to think of the graph to work out what y approaches as x approaches -∞ and +∞)
50 million, 5 million, 500 thousand
Answer:
x = 3.556 or
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation:
(3*x-5)^2-((x+x+25))=0
Evaluate:
=
Pull like factors:
9x^2 - 32x = x • (9x - 32)
x • (9x - 32) = 0
Remember roots of a product:
A product of several terms equals zero. When a product of two or more terms equals zero, then at least one of the terms must be zero. We shall now solve each term = 0 separately. In other words, we are going to solve as many equations as there are terms in the product. Any solution of term = 0 solves product = 0 as well.
Solve : 9x - 32 = 0
Add 32 to both sides of the equation :
9x = 32
Divide both sides of the equation by 9:
x = 32/9 = 3.556
To change from percent to decimal, move the decimal place to the left two times. In this case the decimal is after the 75. so the decimal is .75 To write .75 as a fraction put 75/100 because it is in the hundredths place. Reduce by dividing both parts by 25. the fraction is 3/4
Hope that helped! if you don't get it ask and ill re explain!