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i have been stuck on this problem for days can someone please help me.use
completing the square to describe the graph f(x)=40-12x-x^2.support your answer graphically.
First review what a perfect square is. It is a trinomial which can be obtained by squaring a binomial( multiplying a binomial by itself.
For example a binomial has 2 terms. 2x^2 + 2x
or 4x + 5
When you have a trinomial with an x^2 term and
one x term
and one constant(number)
then the kind of binomial we
are looking to square is of the
form bx + c.
Let's square some binomials.
a) (x + 2) ^2 is (x + 2)(x + 2) = x ^2 + 4x + 4
b) (x + 3 ) ^2 = (x + 3)(x + 3) = x ^2 + 6x + 9
c) (x + 4) ^2 = (x + 4)(x +4) = x^2 + 8x +16
d) AND (x + 5) ^ 2 = (x+5)(x+5) = x^2 +10x + 25
I'm sensing a pattern.
a) 2(2) =4, 2^ 2 =4
b) 3(2) = 6, 3^2 =9
c) 4(2) =8 , 4^2 =8
d) 5(2 = 10, 5^2 =25
Now practice squaring binomials without writing each one twice as aabove.
(x + 6)^2 = X^2 + ___x + ___

For the x coefficient, double the 6.
For the constant term, square the 6
so (x + 6) ^ 2 = x^2 + 12x + 36
x^2 +12x + ? to make a perfect square?
Take 1/2 of 12 and square it . 6^2 = 36.
Make this binomial into a perfect square.
x^2+ 16x + ? ( 1/2 of 16 = 8. 8^2 =64)
x^2 +16x + 64
We cannot have a - in front of the x, because squaring a number
never produces a negative.
Write 40 -12x - x^2 in standard form as
A. -x^2 -12x + 40 = y(or f(x))
Next we replace y with 0.
Why? Because points with y coordinate = 0
are on the x-axis., giving us the
x-intercepts.
Once we know the position of the intercepts we
find the line of symmetry that is
halfway between these points.
A. -x^2 -12x +40 = 0
-x^2 -12x = -40 (Multiply by -1.
B. x^2 + 12x = +40
Now what do we add to left
side to complete the square?
We add 1/2(12) and square it, getting 36. Add to BOTH sides.
B. x^2 +12x +36 = 40 +36
(x + 6)^2 = 76. taking sq. roots:
x + 6 = sq. root of 76
x + 6 = 8.7 0r x + 6 = -8.7
x = 2.7 and x = -14.7
Intercepts are (2.7, 0) and ( -14.7, 0)
Line of symmetry is x = -6
The vertex is on this line .
The x-coordinate is -6, and y coordinate is f(-6)

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Find the intercepts of -x-y=2 and graph the line.
The x-intercept is on the x-axis, the point where the line crosses the x-axis.
Plot a few points on the x-axis and list the coordinates of the points.
Which coordinate is always 0?
For points on the x-axis, the second coordinate is 0, so Y = 0.
In the above equation set y = 0, and solve for x.
-x - y = 2.
-x - 0 = 2
-X = 2
S0 +X = -2
(-2, 0) Is the x-intercept which is on the line.

TO FIND THE Y - INTERCEPT, SET X = 0
-(0) - Y = 2
-Y = 2
SO +Y = -2
AND THE POINT (0, -2) is on the line.
Plot these two points and draw a line through them.
You can also find the slope by calculating the rise over the run.
Remember that each point that is the intercept has 0 for one coordinate. students sometimes get this backwards.
Just remember y-intercept? x = 0
x-intercept ? y = 0

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the system of equations consists of 2 equivalent equations.
you can tell this if you multiply both sides of either equation by -1.

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What do you know about negative exponents?
Do you remember that:
2^(-3) = 1/2^3 so a^(-3) = 1/a^3

4a^(-3) ÷ 8a^(-6)
=( 4)/a^3 ÷8/a^6 = 4/a^3 (a^6/8) = 1/2 a^3

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Judy bought a coat at a 20% discount sale for $72. What was the original price of the coat?
Imagine yourself buying a coat.
The items mentioned are 20% discount sale , original price .
Since we don't use o for a variable (looks like 0)
let's use r for regular price(or original Price)
When something is on sale we subtract something from he regular price to get the sale price.

Fill in the blank: regular price - ______ = sale price.
Regular price - actual discount amt. = sale price
r - d = s
We need to find regular price. That means we are either given the sale price and the discount amt. or we have the info to find them.

There are 2 numbers in the problem.
What does $72 represent. We are told sale price = $72.
Now how do we use 20% to find discount amt.
Discount amt is the small amt taken away from the reg price to get sale price. d = 20% of r so d = .20r


reg. price - discount = sale price
r - d = s
r - .20r = 72
1r - .20r = 72
.80r = 72
I see multiplication , so I divide
r = 72/.80 = $90
Ask yourself if this answer is reasonable.
Is a 20% discount large or small. I'd say small.
If reg price is $90 and sale price is $72 is this a small decrease or a large one. I'd say small.

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The sum of two consecutive odd numbers plus the next even number is 926. Write an algebraic equation that can be used to solve this problem.
We have 3 unknown numbers.
odd number one odd number two and the next even number.
let first odd number be represented by x
The second odd number is x + 2
and the next even number is x + 3

x + x + 2 + x + 3 = 926 *
(first odd ) ( next odd) + ( next even) = 926
combining lime terms we get 3x + 5 = 926
- 5 = -5
__________________
3x = 921

x = 307
Then x + 2 is next odd number = 309
and the next even number is = 310
307 + 309 + 310 = 926

* Since odd numbers are all separated by 2
( for ex 5,7,9,11)
Then if x is first odd number, we have to add 2 to get the
NEXT(consecutive) odd number.