To solve the polynomial equation 9x4 + 2x2 + 7 = 0, we will employ the method of u-substitution. This approach simplifies the equation by reducing its degree.
Step 1: Perform u-substitution
Let u = x2. Therefore, we can rewrite x4 as (x2)2 = u2. Substituting these values into the original equation gives us:
9u2 + 2u + 7 = 0
Step 2: Solve the quadratic equation
Now, we have a standard quadratic equation: 9u2 + 2u + 7 = 0. To find the values of u, we will use the quadratic formula, which is:
u = (-b ± √(b2 – 4ac)) / 2a
For our equation:
- a = 9
- b = 2
- c = 7
Plugging these values into the formula, we get:
u = ( -2 ± √(22 – 4 * 9 * 7) ) / (2 * 9)
u = ( -2 ± √(4 – 252) ) / 18
u = ( -2 ± √(-248) ) / 18
Since we have a negative value under the square root, this indicates that the solutions for u are complex numbers.
Step 3: Simplify the expression
We can express the square root of -248 as:
√(-248) = √(248) * √(-1) = √(4 * 62) * i = 2√62 * i
Thus, the solutions for u become:
u = ( -2 ± 2√62 * i ) / 18
Next, we can simplify this expression:
u = -1/9 ± (√62 * i) / 9
Step 4: Substitute back to find x
Recall that we set u = x2. Hence:
x2 = -1/9 ± (√62 * i) / 9
We can split this into two cases:
- Case 1: x2 = (-1 + √62 * i) / 9
- Case 2: x2 = (-1 – √62 * i) / 9
For each case, we can find the corresponding complex solutions for x by taking the square root:
x = ±√{(-1 ± √62 * i) / 9}
This means that the solutions will be:
- x = ±√{(-1 + √62 * i) / 9}
- x = ±√{(-1 – √62 * i) / 9}
Conclusion:
In summary, the equation 9x4 + 2x2 + 7 = 0 does not have real solutions but instead has complex solutions represented in the forms stated above. U-substitution effectively transformed the polynomial into a more manageable quadratic equation, allowing us to find these complex results.