Quadratic equations

Notice that the position equation has terms involving time and also time squared. Equations that involve a variable squared are called quadratic equations. An equation is said to be quadratic in x if x2 appears as the highest power of x in the equation. Quadratic equations are special for two reasons:
  1. There are special techniques for solving them.
  2. There are two solutions for the squared variable.
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Quadratic equations in math and physics
The variables c, b, and a in the math formula are the coefficients of x0 (= 1), x1 = x, and x2.The two solutions are true for any values of the variables. In physics, we will sometimes need the quadratic solutions above; but in some cases we can also use the easier method of factoring a quadratic equation to solve for the two solutions. Read the text aloud
Solving the problem for an arrow shot upwards
To find the solution we start as usual by defining the initial position as x0 = 0. We are also given that the arrow comes back down to x0 = 0 again 5.0 s after it is fired upward. The position equation becomes a quadratic in time t, from which a t can be factored from each term to give this intermediate equation:
0= v 0 t 1 2 g t 2 0=t ( v 0 gt 2 )
There are two ways in which the result can be zero:
if t=0         or if v 0 gt 2 =0
The solution on the right gives us what we want, which is the value of v0 when t = 5.0 s. The solution on the left is also realistic since it says the arrow was on the ground (x = 0) at the start when t = 0. Read the text aloud

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