![]() Solve for the roots, or the zeroes of quadratic equations. Show you what I'm talking about: it's the quadraticįormula. Things and not know where they came from. Prove it, because I don't want you to just remember Memorize it with the caveat that you also remember how to Videos, you know that I'm not a big fan of memorizing ![]() Really!Įxpose you to what is maybe one of at least the top five They got called "Real" because they were not Imaginary. ![]() NOTE: The Real Numbers did not have a name before Imaginary Numbers were thought of. Meanwhile, try this to get your feet wet: "What's that last bit, complex number and bi" you ask?! The term "imaginary number" now means simply a complex number with a real part equal to 0, that is, a number of the form bi. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. They have some properties that are different from than the numbers you have been working with up to now - and that is it. Well, it is the same with imaginary numbers. ![]() It seemed weird at the time, but now you are comfortable with them. Remember when you first started learning fractions, you encountered some different rules for adding, like the common denominator thing, as well as some other differences than the whole numbers you were used to. They are just extensions of the real numbers, just like rational numbers (fractions) are an extension of the integers. Now we use our algebra skills to solve for "x".Don't let the term "imaginary" get in your way - there is nothing imaginary about them. Total time = time upstream + time downstream = 3 hours (to travel 8 km at 4 km/h takes 8/4 = 2 hours, right?) We can turn those speeds into times using: when going downstream, v = x+2 (its speed is increased by 2 km/h).when going upstream, v = x−2 (its speed is reduced by 2 km/h).Let v = the speed relative to the land (km/h)īecause the river flows downstream at 2 km/h:.Let x = the boat's speed in the water (km/h).There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land: What is the boat's speed and how long was the upstream journey? The negative value of x make no sense, so the answer is:Įxample: River Cruise A 3 hour river cruise goes 15 km upstream and then back again. The desired area of 28 is shown as a horizontal line. There are many ways to solve it, here we will factor it using the "Find two numbers that multiply to give ac, and add to give b" method in Factoring Quadratics: It looks even better when we multiply all terms by −1: (Note for the enthusiastic: the -5t 2 is simplified from -(½)at 2 with a=9.8 m/s 2)Īdd them up and the height h at any time t is:Īnd the ball will hit the ground when the height is zero: Gravity pulls it down, changing its position by about 5 m per second squared: It travels upwards at 14 meters per second (14 m/s): (Note: t is time in seconds) The height starts at 3 m: Ignoring air resistance, we can work out its height by adding up these three things:
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