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Identify all the real roots of 4x^4+31x^3-4x^2-89x+22=0
4 months ago
Q:
Identify all the real roots of 4x^4+31x^3-4x^2-89x+22=0
Accepted Solution
A:
Solve for x over the real numbers:
4 x^4 + 31 x^3 - 4 x^2 - 89 x + 22 = 0
The left hand side factors into a product with three terms:
(x + 2) (4 x - 1) (x^2 + 6 x - 11) = 0
Split into three equations:
x + 2 = 0 or 4 x - 1 = 0 or x^2 + 6 x - 11 = 0
Subtract 2 from both sides:
x = -2 or 4 x - 1 = 0 or x^2 + 6 x - 11 = 0
Add 1 to both sides:
x = -2 or 4 x = 1 or x^2 + 6 x - 11 = 0
Divide both sides by 4:
x = -2 or x = 1/4 or x^2 + 6 x - 11 = 0
Add 11 to both sides:
x = -2 or x = 1/4 or x^2 + 6 x = 11
Add 9 to both sides:
x = -2 or x = 1/4 or x^2 + 6 x + 9 = 20
Write the left hand side as a square:
x = -2 or x = 1/4 or (x + 3)^2 = 20
Take the square root of both sides:
x = -2 or x = 1/4 or x + 3 = 2 sqrt(5) or x + 3 = -2 sqrt(5)
Subtract 3 from both sides:
x = -2 or x = 1/4 or x = 2 sqrt(5) - 3 or x + 3 = -2 sqrt(5)
Subtract 3 from both sides:
Answer:Β x = -2 or x = 1/4 or x = 2 sqrt(5) - 3 or x = -3 - 2 sqrt(5)