How many real roots does the equation (9x^5)-(4x^4)+(8x^3)-4x+1=0 have?

  • Thread starter wildcat12
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In summary, to show that the equation (9x^5)-(4x^4)+(8x^3)-4x+1=0 has exactly 3 real roots, one can use the intermediate value theorem and plot the graph to approximate the roots. It is also important to show that there is only one inflection point, indicating 3 roots instead of 5.
  • #1
wildcat12
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Homework Statement



Show that the equation (9x^5)-(4x^4)+(8x^3)-4x+1=0 has exactly 3 real roots.

Homework Equations





The Attempt at a Solution


I found a derivative of (45x^4)-(16x^3)+(24x^2)-4. I am not sure how to set this equal to zero
 
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  • #2
Use the intermediate value theorem, if f(x) is a continuous function and there exist point a<b such that f(a)<0<f(b) or f(a)>0>f(b) then there is a point c, where f(c)=0.

My advice is to plot the graph and find approximatly where they are and then use the above to prove it so.
 
  • #3
Also, you should show there's only one inflection point, so you have only 3 roots and not 5.
 

What is the process of finding roots?

The process of finding roots involves solving an equation to determine the value or values that make the equation true. These values are known as roots or solutions.

Why is finding roots important in science?

Finding roots is important in science because it allows us to solve equations and understand the relationships between different variables. This is crucial in fields such as physics, chemistry, and engineering.

What are the different methods for finding roots?

There are several methods for finding roots, including the quadratic formula, factoring, and using graphs. Other methods include the bisection method, Newton's method, and the secant method.

What are complex roots?

Complex roots are solutions to equations that involve imaginary numbers. These numbers are expressed as a combination of a real number and the imaginary unit, i (equal to the square root of -1).

How do you know if an equation has no real roots?

An equation has no real roots if the solutions involve complex numbers. This can be determined by looking at the discriminant (b^2-4ac) of a quadratic equation. If the discriminant is negative, the equation has no real roots.

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