Show that the equation x^4/2021 − 2021x^2 − x − 3 = 0 has at least two real roots.
1 answer:
The roots of an equation are simply the x-intercepts of the equation.
See below for the proof that has at least two real roots
The equation is given as:
There are several ways to show that an equation has real roots, one of these ways is by using graphs.
See attachment for the graph of
Next, we count the x-intercepts of the graph (i.e. the points where the equation crosses the x-axis)
From the attached graph, we can see that crosses the x-axis at approximately <em>-2000 and 2000 </em>between the domain -2500 and 2500
This means that has at least two real roots
Read more about roots of an equation at:
You might be interested in
10/8 in simplest form:
First, we can start off by finding the GCF of the denominator and the numerator. To do so, we need to list the factors of each of them and find the common ones.
Factors of 10: 1, 2, 5, 10
Factors of 8: 1, 2, 4, 8
We can see that our common factors are 1 and 2, considering that both the numerator and denominator has the same factors (1 and 2). Since we have our common factors, we need to find the greatest. Which is the greatest out of 1 and 2? The GCF is 2 because it is bigger than 1.
Second, divide the numerator and the denominator by the GCF (2).
Third, now we can revise our fraction and turn it into the simplest form. Take the two numbers you just got above us and put them in their numerator and denominator spot. You should get:
Answer in fraction form:
Answer in decimal form:
Answer in mixed number form:
First, we can start off by finding the GCF of the denominator and the numerator. To do so, we need to list the factors of each of them and find the common ones.
Factors of 10: 1, 2, 5, 10
Factors of 8: 1, 2, 4, 8
We can see that our common factors are 1 and 2, considering that both the numerator and denominator has the same factors (1 and 2). Since we have our common factors, we need to find the greatest. Which is the greatest out of 1 and 2? The GCF is 2 because it is bigger than 1.
Second, divide the numerator and the denominator by the GCF (2).
Third, now we can revise our fraction and turn it into the simplest form. Take the two numbers you just got above us and put them in their numerator and denominator spot. You should get:
Answer in fraction form:
Answer in decimal form:
Answer in mixed number form:
Answer:
(27.55, 7.22), (-11.3, 3.21).
Step-by-step explanation:
When is the tangent to the curve horizontal?
The tangent curve is horizontal when the derivative is zero.
The derivative is:
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
.
This polynomial has roots such that
, given by the following formulas:
In this question:
So
Then
So
Enter your answers as a comma-separated list of ordered pairs.
We found values of t, now we have to replace in the equations for x and y.
t = 3.35
The first point is (27.55, 7.22)
t = -2.685
The second point is (-11.3, 3.21).