Answer:
Explanation:
To solve the equation 9x^4 – 2x^2 – 7 = 0 using u substitution, we can let u = x^2.
Substituting u into the equation, we get 9u^2 – 2u – 7 = 0.
This is a quadratic equation in terms of u. To solve for u, we can use the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / (2a), where a = 9, b = -2, and c = -7.
Plugging in these values, we have:
u = (-(-2) ± √((-2)^2 - 4(9)(-7))) / (2(9))
Simplifying further:
u = (2 ± √(4 + 252)) / 18
u = (2 ± √256) / 18
u = (2 ± 16) / 18
This gives us two possible values for u:
u = (2 + 16) / 18 = 18 / 18 = 1
u = (2 - 16) / 18 = -14 / 18 = -7/9
Now that we have the values of u, we can substitute them back into the equation u = x^2:
For u = 1, we have x^2 = 1, which gives us two solutions:
x = √1 = 1
x = -√1 = -1
For u = -7/9, we have x^2 = -7/9. However, this equation has no real solutions since the square of a real number cannot be negative.
Therefore, the solutions to the equation 9x^4 – 2x^2 – 7 = 0 are:
x = 1
x = -1