Question

(1-2x)4 - 5(1-2x)2= -4

Answer 1

4-8х-10+20х=-4
12х-6=-4
12х=-4+6
12х=2
х=1/6

Answer 2

To solve the equation (1-2x)^4 - 5(1-2x)^2 = -4, we can start by simplifying the expression using the properties of exponents. Let's break it down step by step:

Let's denote (1-2x) as a variable, y, for simplicity. So we have: y^4 - 5y^2 = -4

Now, let's rearrange the equation to bring all the terms to one side: y^4 - 5y^2 + 4 = 0

This is a quadratic equation in terms of y^2. To solve it, we can substitute z = y^2: z^2 - 5z + 4 = 0

Now we can factor this quadratic equation: (z - 4)(z - 1) = 0

Setting each factor equal to zero, we have two possibilities: z - 4 = 0 or z - 1 = 0

For the first case: z - 4 = 0 y^2 - 4 = 0 (y - 2)(y + 2) = 0

So we have two solutions: y - 2 = 0 or y + 2 = 0 y = 2 or y = -2

For the second case: z - 1 = 0 y^2 - 1 = 0 (y - 1)(y + 1) = 0

So we have two additional solutions: y - 1 = 0 or y + 1 = 0 y = 1 or y = -1

Therefore, the equation has four possible solutions: y = 2, -2, 1, -1

Now, we substitute back y = (1-2x): 1 - 2x = 2 or 1 - 2x = -2 or 1 - 2x = 1 or 1 - 2x = -1

Solving each equation for x, we get: -2x = 1 or -2x = 3 or -2x = 0 or -2x = 2 x = -1/2 or x = -3/2 or x = 0 or x = -1

So the solutions to the original equation are: x = -1/2, -3/2, 0, -1