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Aug 17, 2021 · From The Factor Theorem, Theorem \(\PageIndex{3}\), we can get yet another insight into the problems associated with solving polynomial equations; that is, finding the zeros of a polynomial. The initial important idea here is that the zero \(a\) is from the ground field \(F\text{.}\)
- Balls, Arrows, Missiles and Stones
- Example: Throwing A Ball
- Example: New Sports Bike
- Example: Small Steel Frame
- Example: River Cruise
- Example: Resistors in Parallel
- Others
When you throw a ball (or shoot an arrow, fire a missile or throw a stone) it goes up into the air, slowing as it travels, then comes down again faster and faster ... ... and a Quadratic Equationtells you its position at all times!
A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?
Add them up and the height h at any time tis: h = 3 + 14t − 5t2 And the ball will hit the ground when the height is zero: 3 + 14t − 5t2= 0 Which is a Quadratic Equation! In "Standard Form" it looks like: −5t2+ 14t + 3 = 0 It looks even better when we multiply all terms by −1: 5t2− 14t − 3 = 0 Let us solve it ... 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: ac = −15, and b = −14....
You have designed a new style of sports bicycle! Now you want to make lots of them and sell them for profit. Your costsare going to be: 1. $700,000 for manufacturing set-up costs, advertising, etc 2. $110 to make each bike Based on similar bikes, you can expect salesto follow this "Demand Curve": Where "P" is the price. For example, if you set the ...
Your company is going to make frames as part of a new product they are launching. The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm2 The inside of the frame has to be11 cm by 6 cm What should the width xof the metal be? Area of steel before cutting: Area of steel after cutting out the 11 × 6 ...
A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?
There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land: 1. Let x= the boat's speed in the water (km/h) 2. Let v= the speed relative to the land (km/h) Because the river flows downstream at 2 km/h: 1. when going upstream, v = x−2(its speed is reduced by 2 km/h) 2. when going downstream, v = x+2(its speed is increased by 2 km/h) We can turn those speeds into times using: time = distance / speed (to travel 8 km at 4 km/h takes 8/4 = 2 hours...
Two resistors are in parallel, like in this diagram: The total resistance has been measured at 2 Ohms, and one of the resistors is known to be 3 ohms more than the other. What are the values of the two resistors? The formula to work out total resistance "RT" is: 1RT = 1R1 + 1R2 In this case, we have RT = 2 and R2 = R1+ 3 1 2 = 1 R1 + 1 R1+3 To get ...
Quadratic Equations are useful in many other areas: For a parabolic mirror, a reflecting telescope or a satellite dish, the shape is defined by a quadratic equation. Quadratic equations are also needed when studying lenses and curved mirrors. And many questions involving time, distance and speed need quadratic equations.
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Oct 1, 2024 · This equation is true only when one of the factors on the left side is zero. Each of the factors on the left side is zero only when x is a number between 1 and 9; for example, (x-1)=0 when x=1.
- Alan Veliz-Cuba
The projectile-motion equation is s(t) = −½ gx2 + v0x + h0, where g is the constant of gravity, v0 is the initial velocity (that is, the velocity at time t = 0), and h0 is the initial height of the object (that is, the height at of the object at t = 0, the time of release).
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If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero. If the graph touches the x -axis and bounces off of the axis, it is a zero with even multiplicity. If the graph crosses the x -axis at a zero, it is a zero with odd multiplicity.
