This is an example of an arithmetic series: 6 + 8 + 10 + 12 + . . . + (4 + 2n) + . . .
1 answer:
True. A series is arithmetic if its terms come from an arithmetic sequence.
And a sequence is said to be arithmetic any of its two consecutive terms differ by a fixed difference, common to any consecutive pair.
As you can see, in this case, all terms differ by 2, so the sequence
is arithmetic, and thus the series
is arithmetic as well.
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You can find “hypotenuse” using Pythagorean to get the length of one of those steps, 12.5, and you get 150 when you multiply that number by 12 for each of the 12 steps
9514 1404 393
Explanation:
1) 2x^2 -5x -3 = 0 . . . . standard form equation
To convert this to factored form, you can look for factors of the product (2)(-3) that have a sum of -5. It can help to start by listing the ways that -6 can be factored. Since we want the sum of factors to be negative, we want to have larger negative factors.
-6 = (1)(-6) = (2)(-3)
The sums of these factor pairs are -5 (what we want) and -1 (not relevant). We can call these factors p=1 and q=-6.
If a = 2 is the leading coefficient of our standard form quadratic, we want to use these factors in the form ...
(ax +p)(ax +q)/a . . . . . factored form of the quadratic
(2x +1)(2x +(-6))/2 . . . .fill in the values we know
(2x +1)(x -3) . . . . . . . factor 2 from the second binomial
So, the factored form of the quadratic equation is ...
(2x +1)(x -3) = 0 . . . . factored form equation
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2) f(x) = x^2 +7x +10 . . . . standard form quadratic function
Using the thinking process described above, we are looking for factors of 10 that have a sum of 7. We know those are 2 and 5. So, the factored form of the function is ...
f(x) = (x +2)(x +5) . . . . . . factored form quadratic function
The leading coefficient is 1, so we have no further work to do.
<u>Roots, x-intercepts, zeros</u>
The graph attached below shows this function crosses the x-axis when x=-2 and x = -5. These values of x are variously called "roots", "x-intercepts", and "zeros" of the function. They are values for which the factors and the function are zero. (x+2=0 when x=-2, for example)
<u>Solutions</u>
Often, we are interested in solving the equation ...
f(x) = 0
For that equation, the <em>solutions</em> <u>are</u> the <em>zeros</em> or <em>x-intercepts</em> or <em>roots</em>. The graph attached also shows solutions for ...
f(x) = 4
Those solutions are x = -6 and x = -1. The function value is not zero for these values of x, so the <em>roots</em>, <em>x-intercepts</em>, or <em>zeros</em> <u>are not</u> <em>solutions</em> to this equation.
