MATH SOLVE

5 months ago

Q:
# What are the solutions of the equation 9x^4 – 2x^2 – 7 = 0? Use u substitution to solve.

Accepted Solution

A:

The answers are x = -1, 1, i√7/3, -i√7/3.

Solution:

Solving by making a u-substitution, we let u = x² and rewrite the equation in quadratic form.

9u² - 2u - 7 = 0

We can now solve the quadratic equation by factoring. We need two numbers whose sum is -2 and whose product is -7. In this case, it would have to be 7 and -1, considering the term 9u². Hence, we can also write our equation in the factored form

(u - 1)(9u + 7) = 0

Now we have a product of two expressions that is equal to zero, which means any u value that makes either (u - 1) or (9u + 7) zero will make their product zero.

u - 1 = 0 => u = 1

9u + 7 = 0 => u = -7/9

We substitute back x² = u to calculate for x.

u = 1 => x² = 1 => x = -1, 1

u = -7/9 => x² = -7/9 => x = i√7/3, -i√7/3

Therefore, the solutions are −1, 1, i√7/3, and -i√7/3.

Solution:

Solving by making a u-substitution, we let u = x² and rewrite the equation in quadratic form.

9u² - 2u - 7 = 0

We can now solve the quadratic equation by factoring. We need two numbers whose sum is -2 and whose product is -7. In this case, it would have to be 7 and -1, considering the term 9u². Hence, we can also write our equation in the factored form

(u - 1)(9u + 7) = 0

Now we have a product of two expressions that is equal to zero, which means any u value that makes either (u - 1) or (9u + 7) zero will make their product zero.

u - 1 = 0 => u = 1

9u + 7 = 0 => u = -7/9

We substitute back x² = u to calculate for x.

u = 1 => x² = 1 => x = -1, 1

u = -7/9 => x² = -7/9 => x = i√7/3, -i√7/3

Therefore, the solutions are −1, 1, i√7/3, and -i√7/3.