x^4+(100/(x^4+1))=19 Multiply all terms by x^4+1: x^4(x^4+1)+100=19(x^4+1) Distribute: x^8+x^4+100=19x^4+19 Combine like terms: x^8=18x^4-81 Take fourth root of all terms: x^2=2.06x-3 Simplify: x^2-2.06x+3=0 Apply quadratic formula: x=1.03+1.39i, 1.03-1.39i I’m not seeing what I did wrong
Your mistake is on the step where you take the fourth root, root is not linear so root(a+b) != root(a) + root(b) I am not sure this is a good thing to use the fourth root, a better way is to go back to your x⁸ = 18x⁴ -81, take all members on the same side : x⁸-18x⁴+81 = 0 Then you can rewrite this in the form of (a - b)² : (x⁴)² - 2 * 9 * x⁴ + 9² = 0 (x⁴ - 9)² = 0 And the solutions come easily by solving x⁴ = 9 x² = 3 or -3 x = sqrt(3) or -sqrt(3) or sqrt(3)i or -sqrt(3)i (or only the two first if you solve this for real numbers)
x^4+1+100/(x^4+1)=20 u=x^4+1==>u^2-20x+100=0 (u-10)=0 ==> u=10 So x^4=9 x=sqr(3) or x=-sqr(3) or x=sqr(3)*i or x=-sqr(3)*i all with multiplicity 2. Easy question.
Nothing frightening! Easy peasy!
Great video
Was soll der Begriff 'Einschaltungszeichen?
x^4+(100/(x^4+1))=19
Multiply all terms by x^4+1:
x^4(x^4+1)+100=19(x^4+1)
Distribute:
x^8+x^4+100=19x^4+19
Combine like terms:
x^8=18x^4-81
Take fourth root of all terms:
x^2=2.06x-3
Simplify:
x^2-2.06x+3=0
Apply quadratic formula:
x=1.03+1.39i, 1.03-1.39i
I’m not seeing what I did wrong
Your mistake is on the step where you take the fourth root, root is not linear so root(a+b) != root(a) + root(b)
I am not sure this is a good thing to use the fourth root, a better way is to go back to your x⁸ = 18x⁴ -81, take all members on the same side : x⁸-18x⁴+81 = 0
Then you can rewrite this in the form of (a - b)² : (x⁴)² - 2 * 9 * x⁴ + 9² = 0 (x⁴ - 9)² = 0
And the solutions come easily by solving x⁴ = 9 x² = 3 or -3 x = sqrt(3) or -sqrt(3) or sqrt(3)i or -sqrt(3)i (or only the two first if you solve this for real numbers)
x^4+1+100/(x^4+1)=20
u=x^4+1==>u^2-20x+100=0
(u-10)=0 ==> u=10
So x^4=9
x=sqr(3) or x=-sqr(3) or x=sqr(3)*i or x=-sqr(3)*i all with multiplicity 2.
Easy question.
x = 1,3
No, neither of those answers is correct.
x⁴ + 1 = u
u + 100/u = 20
u² - 20u + 100 = 0
u = 10
x⁴ + 1 = 10
x⁴ = 9
x² = ± 3
*x = ± √3* ∨ *x = ± i√3*
another way
x⁴ + 1 + 100/(x⁴ + 1) = 20
(x⁴ + 1)/10 + 10/(x⁴ + 1) = 2
(x⁴ + 1)/10 = u
u + 1/u = 2
u² - 2u + 1 = 0
(u - 1)² = 0 => u = 1
(x⁴ + 1)/10 = 1 => x⁴ = 9
*x = ± √3* ∨ *x = ± i√3*