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Sure, one is a square of one itself and also of -1:

1=1 x 1

1= -1 x -1

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Let's say a problem asks you to compute a length of something, or some other positive number. And let's say that the problem is expressed as a quadratic equation with one solution positive and one negative. In this instance, the negative solution is unacceptable.

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Good question. A graph of a quadratic equation is a parabola, and because it has two wings pointing inthe same direction, a parabola could cross the x axis twice. By the way, cubic equations can have up to 3 different solution, quartic (4th power) equations can have up to 4 solutions, etc.

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Airline's revenue is: Number of passengers * ticket price.

Number of passengers also depends on ticket price:

Number of passengers = 2,000 - 50 * (ticket price - 100)/10

So, airline's revenue is:

(2,000 - 50 * (ticket price - 100)/10)*ticket price

Denote ticket price as x

Revenue = (2000-50*(x-100)/10)*x

When you simplify it, you get a quadratic expression, that you can plot on my site.

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That's a tough problem for an 8 year old, I am rather impressed. After 26 years, the brother of your cousin will be 12+26 years old, and your sister will be 14 + 26. So, your cousin will be 38 and your sister will be 40 (which is VERY old). The sum of their ages will be 78, half of that sum will be 39.

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We have length = width + 4. The area is length*width and is 140. Because length = width+4, we have (width+4)*width = 140, or . That's a simple quadratic equation that you can solve on my website. (10 seems to be the obvious root)

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If you fill a 9 cup container, and then pour off to fill a full 4 cup
container, dump water from the 4 cup container and fill it again, you
would be left with 9-4-4 = one cup of water in the big container. Pour
that one cup of water into the 4 cup container and set it aside. Fill
the 9 cup again, pour off 3 cups into the 4 cup container (which
already holds one cup, so it has space for 3 more cups), you will end
up with 6 cups in the big container.

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Your task here is mainly to write an equation. Write it in words first:

number PLUS same number PLUS same 1.25 * number = 2400.

Replace number with x:

x + x + 1.25*x = 2400

3.25*x = 2400

this equation does not solve for a natural number x, the result is x = 738.461..., which makes me suspect that you either mis-stated the problem or I did not understand it.

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there is a calculator for this on my site.

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This is a great question. When you draw pie charts, you deal with numbers as percentages of the total. So, let's say some number is 10% of the total (0.1, or one tenth). What angle should it require in a pie chart? There are 360 degrees in the pie. The angle should be 0.1 times 360, or 36 degrees. This way, just as all numbers in a total add to 100% of the total, all angles in the pie chart will also add up to 360 degrees.

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You should ask two questions separately, but here goes.

Denote the amount invested at 7% as x dollars. The remaining amount to be invested is, obviously, 2000-x. Amount of interest at 7% is x*0.07. Amount of interest invested at 8% is (2000-x)*0.08. So, we have x*0.07+(2000-x)*0.08 = 154.

x*0.07+2000*0.08 - x*0.08 = 154

2000*0.08-154 + x*0.07 - x*0.08 = 0

160-154-0.01x = 0

6 = 0.01x

x = 600, 2000-x = 1400

Answer: He invested $600 at 7% and $1,400 at 8%.

Regarding your unhealthy interest in alcohol solutions :), I have a word problem solver for exactly this problem. Go to word problems from my home page.

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43.7% increase means that the number of employees this year is 1.437 times the number of employees last year. That means that the number of employees last year was times the number of employees this year, or , or 1633744 employee.

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A percentage is a ratio of one number to another, expressed in 's. So, if the first number is bigger than another, then the percentage would be above 100. For example, if your car can go 60 mph and you expected it to go only 50 mph, then it will go at 120% the expected speed.

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If you are studying complex numbers (and know what is an imaginary number i), then we can write exponents of e that give negative results. If you are studying basic logarithms and exponents and do not know what i is, then the above equation makes no sense.

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My guess is that f-1(7) is the inverse function and -1 is not an exponent but a sign of inverse function. If so, then f-1(7) = 3.

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What this really is is a quadratic equation. Note that (see lesson on logarithms). So, replace with y. You will have: . Solve it for y, you can use my quadratic solver on this website which will show work. Once you get y, solve equation , which means take third root of y.

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A logarithm is only defined for positive numbers. The reason is that for most negative numbers, powering then to, say, degree is impossible. Power of 1/2 means square root and, as we know, there is no real number representing a square root of a negative. So, non-integer powers and logarithms are defined only for positive numbers.