Find all positive solutions to:
x^6 + 7x = 4x^4 - x^2 + 8
x^6 - 4x^4 + x^2 + 7x - 8 = 0
Wolfram said there is a positive solution at
appr. x = 1.80445
http://www.wolframalpha.com/input/?i=x%5E6+-+4x%5E4+%2B+x%5E2+%2B+7x+-+8+%3D+0
what a mess, this would be horrible without technology .
I won't have access to this on a test. How can I work it on a calculator.
To find all positive solutions to the equation x^6 + 7x = 4x^4 - x^2 + 8, we need to manipulate the equation and solve for x.
1. Rearrange the equation: x^6 + 7x - (4x^4 - x^2 + 8) = 0.
2. Combine like terms: x^6 + 7x - 4x^4 + x^2 - 8 = 0.
3. Write the equation in descending powers of x: x^6 - 4x^4 + x^2 + 7x - 8 = 0.
4. Factor the equation: We can try factoring by grouping. Notice that the equation has five terms. Arrange the terms in pairs and factor them separately.
(x^6 - 4x^4) + (x^2 + 7x - 8) = 0.
Factor out the common factor from the first group: x^4(x^2 - 4).
For the second group, we need to factorize the quadratic expression: x^2 + 7x - 8.
Since the quadratic expression does not factor nicely, we can use the quadratic formula to find its roots.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / 2a.
In this case, a = 1, b = 7, and c = -8.
Applying the quadratic formula, we have:
x = (-7 ± √(7^2 - 4(1)(-8))) / 2(1).
Simplifying further, we have:
x = (-7 ± √(49 + 32)) / 2.
x = (-7 ± √(81)) / 2.
x = (-7 ± 9) / 2.
Now, we have two possible values for x:
x1 = (-7 + 9) / 2 = 1.
x2 = (-7 - 9) / 2 = -8.
5. Check for additional solutions: We found two possible values for x, but we are only interested in positive solutions. Since x2 = -8 is not positive, we ignore it.
Therefore, the only positive solution to the equation is x = 1.