Find the differential equation of dy/dx=(3x+y+4)^2
2 answers:
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Answer: x = -2±√2 i or no real roots. See the explanation below.
Step-by-step explanation:
If you draw the graph, you will notice that there are no common points or you can say that there are no x-intercepts.
This means that the equation doesn't have any real roots.
If there are 2 common points (2 x-intercepts) then there are 2 real roots (answers.)
If there is only one common point (vertex is on x-axis) then there is only 1 real root.
If there aren't any common points (no x-intercepts) then there are no real roots.
So first, I'm gonna convert the equation into function and convert the standard form to vertex form.
We notice that the vertex is at (-2,2), the graph doesn't have x-intercepts (Graph below)
So the answer is clearly complex.
If you solve this by solving the equation then it'll be.
The answer is complex.
However, you can find the value of x easily by graphing. For example.
Factor then you get
Or you can convert it to vertex form then graph.
The vertex is at (-3,-4) then graph parabola (x-intercepts determine the roots of quadratic equation. For the graph, parabola intercepts x at (-5,0) and (-1,0) that means both -5 and -1 both are the value of x.
Note:
From the standard form, The graph is called Parabola, and it's Quadratic Function.
a is how the graph is, for example. If a>0, the graph is supine parabola and if a<0 then the graph is inverted parabola.
|a| is determined if the graph is wide or narrow, the more the value of a is, the more narrow it will be. The less the value of a is, the more wide it will be.
b is what determined the change of graph (likely vertex). Without b, then it'd be which only move y-axis and not x-axis.
c is what determined the y-intercept and y-axis. If there is no b or the value of b is 0 then the vertex is at (0,c). If there is the value of b then c is only determined the y-intercept.
