Solve x^4-4x^3+4x^2-9=0 given that 1+isqrt2 is a root.
I need all of the steps.
Thanks
x^4-4x^3+4x^2-9=0
Given 1+i√2 is a root, then
1-i√2 is also a root (conjugate).
The product of the two roots gives a real factor of
(x-1-i√2)(x-1+i√2)
=x²-2x+3
Do a long division of
(x^4-4*x^3+4*x^2-9)÷(x²-2x+3)
=x²-2x-3
and solve by factoring to get
(x-3)(x+1)=0
Alternatively,
note that
x^4-4x^3+4x^2 is a perfect square
=(x²-2x)²
so the left-hand-side can be factorized:
(x²-2x)²-3²=0
or
(x²-2x+3)(x²-2x-3)=0
which can be further factorized as:
(x²-2x+3)(x-3)(x+1)=0
Solve the first factor by the quadratic formula to get the conjugate complex roots, and the remainder for x=3, x=-1.
To solve the given equation x^4 - 4x^3 + 4x^2 - 9 = 0 using the given information that 1 + i√2 is a root, we can follow the steps below:
Step 1: As 1 + i√2 is a root, its conjugate 1 - i√2 must also be a root.
Step 2: Express the given equation in terms of the roots:
(x - (1 + i√2))(x - (1 - i√2)) = 0
Step 3: Simplify the equation by expanding the product:
(x - 1 - i√2)(x - 1 + i√2) = 0
[(x - 1)^2 - (i√2)^2] = 0
(x - 1)^2 - (i√2)^2 = 0
(x - 1)^2 + 2 = 0
Step 4: Apply the difference of squares identity:
(x - 1 - √2i)(x - 1 + √2i) = 0
Step 5: Set each factor equal to zero and solve for x:
x - 1 - √2i = 0 and x - 1 + √2i = 0
For the first factor:
x - 1 - √2i = 0
Move -1 to the right side:
x = 1 + √2i
For the second factor:
x - 1 + √2i = 0
Move -1 to the right side:
x = 1 - √2i
So, the two additional roots are x = 1 + √2i and x = 1 - √2i.
Hence, the four roots of the equation x^4 - 4x^3 + 4x^2 - 9 = 0 are:
x = 1 + i√2, x = 1 - i√2, x = 1 + √2i, and x = 1 - √2i.
To solve the equation x^4 - 4x^3 + 4x^2 - 9 = 0, given that 1 + i√2 is a root, you can use the fact that complex roots of polynomial equations always appear as conjugate pairs.
Here are the steps to solve the equation:
Step 1: Write down the given root and its conjugate.
Given root: 1 + i√2
Conjugate root: 1 - i√2
Step 2: Use the given roots to form two linear factors.
Factors: (x - (1 + i√2)) and (x - (1 - i√2))
Step 3: Expand the factors.
(x - (1 + i√2)) = x - 1 - i√2
(x - (1 - i√2)) = x - 1 + i√2
Step 4: Multiply the two factors together.
Multiplying the factors, we get: (x - 1 - i√2)(x - 1 + i√2)
Step 5: Simplify the expression obtained in Step 4.
Expanding the expression, we have: (x - 1)^2 - (i√2)^2
The expression simplifies further as: (x - 1)^2 - (i^2)(√2)^2
Since i^2 = -1, we can rewrite it as: (x - 1)^2 + 2
Step 6: Set the equation equal to zero.
Now, set the simplified expression obtained in Step 5 equal to zero:
(x - 1)^2 + 2 = 0
Step 7: Solve for x.
(x - 1)^2 = -2
Taking the square root of both sides, we get:
x - 1 = ±√(-2)
Since we are dealing with complex numbers, we can express the square root of -2 as ±i√2.
x - 1 = ±i√2
Adding 1 to both sides, we have:
x = 1 ± i√2
Therefore, the equation x^4 - 4x^3 + 4x^2 - 9 = 0 has two complex roots: 1 + i√2 and 1 - i√2, and their conjugates 1 - i√2 and 1 + i√2, respectively.