Question

Solve the equation after making an appropriate substitution. (11x-2)² -2(11x-2)-8=0 Determine the appropriate substitution using the new variable u. U = a) u = 11x - 2 b) u = 11x + 2 c) u = 11x d) u = 2(11x - 2)

Answer 1

To solve the equation (11x - 2)² - 2(11x - 2) - 8 = 0 using an appropriate substitution, we can let u = 11x - 2. So, the correct choice for the substitution is:

a) u = 11x - 2

Now, we can make this substitution in the equation:

(u)² - 2(u) - 8 = 0

Now, we have a quadratic equation in terms of u:

u² - 2u - 8 = 0

We can solve this quadratic equation for u. To do so, you can use the quadratic formula:

u = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -2, and c = -8. Plugging these values into the quadratic formula:

u = (2 ± √((-2)² - 4(1)(-8))) / (2(1))

Now, calculate the values under the square root:

u = (2 ± √(4 + 32)) / 2

u = (2 ± √36) / 2

u = (2 ± 6) / 2

Now, you can find the two possible solutions for u:

1. u₁ = (2 + 6) / 2 = 8 / 2 = 4

2. u₂ = (2 - 6) / 2 = -4 / 2 = -2

Now that we have found the values of u, we need to find the corresponding values of x. Recall that u = 11x - 2:

1. For u₁ = 4:

4 = 11x - 2

Add 2 to both sides:

4 + 2 = 11x

6 = 11x

Divide by 11:

x₁ = 6/11

2. For u₂ = -2:

-2 = 11x - 2

Add 2 to both sides:

-2 + 2 = 11x

0 = 11x

Divide by 11:

x₂ = 0

So, the solutions for the original equation (11x - 2)² - 2(11x - 2) - 8 = 0 are:

x₁ = 6/11

x₂ = 0

Explanation: