I asked this question previously, but I was told you were unable to get the answer with rational numbers. I asked my teacher, and he said we could use estimates.

Question

I asked this question previously, but I was told you were unable to get the answer with rational numbers. I asked my teacher, and he said we could use estimates.

y=x^4-4x^3+16
y=(x-2)(x^3-2x^2-4x-8)
so i used 3.68

and i got this (x-2)(x-3.68)(x^2+1.68x+2.1824)

Could someone help me finish?

Answer 1

The remaining quadratic has no solution in the set of real numbers.

The value of b^2 - 4ac in the formula is negative, so we cannot take the √ of a negative number.

your only real solutions are x = 2 and x = appr. 3.68

Answer 2

When you first asked this question, you asked for the factors, you did not ask for a solution to the corresponding equation x^4-4x^3+16 = 0

That is how it was answered.

Answer 3

Certainly! I'd be happy to help you finish solving the equation.

You've correctly factored the quadratic expression (x^2 + 1.68x + 2.1824) as one of the factors of the given polynomial equation (y = (x-2)(x-3.68)(x^2+1.68x+2.1824)).

Now, we can set each factor equal to zero and solve for x. Let's start with the first factor (x - 2) and set it equal to zero:

x - 2 = 0

By adding 2 to both sides of the equation, we get:

x = 2

So, the value of x for the first factor is 2.

Now, let's move on to the second factor (x - 3.68) and set it equal to zero:

x - 3.68 = 0

By adding 3.68 to both sides of the equation, we get:

x = 3.68

So, the value of x for the second factor is 3.68.

Lastly, let's focus on the quadratic factor (x^2 + 1.68x + 2.1824). This factor cannot be easily factored further using rational numbers. However, you mentioned that your teacher allowed you to use estimates. In this case, you can use the quadratic formula to find the approximate values of x.

The quadratic formula is given by:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our quadratic factor (x^2 + 1.68x + 2.1824), we have:

a = 1
b = 1.68
c = 2.1824

Substituting these values into the quadratic formula, we get:

x = (-1.68 ± sqrt((1.68)^2 - 4(1)(2.1824))) / (2(1))

Simplifying further:

x = (-1.68 ± sqrt(1.5104 - 8.7296)) / 2
x = (-1.68 ± sqrt(-7.2192)) / 2

At this point, we notice that the value inside the square root is negative, indicating that there are no real solutions (rational or approximate) for this quadratic factor. Therefore, we cannot find any additional values of x from this factor.

In conclusion, the solutions for the equation y = (x-2)(x-3.68)(x^2+1.68x+2.1824) are x = 2 and x = 3.68.