Question

(x²-2x-5)²-5(x²-2x-5)=6

Answer 1

Ответ:

x1= 1+√5

x2= 1-√5

x3= 1+2√3

x4= 1-2√3

Answer 2

Let's solve the given equation step by step. The equation is:

(x² - 2x - 5)² - 5(x² - 2x - 5) = 6

Step 1: Let's simplify the equation by introducing a variable to make the substitution more manageable. Let's set a new variable u = (x² - 2x - 5):

So the equation becomes:

u² - 5u = 6

Step 2: Now, let's rewrite the equation as a quadratic equation:

u² - 5u - 6 = 0

Step 3: Factor the quadratic equation (if possible):

To factor the equation, we need two numbers whose product is -6 and whose sum is -5.

The numbers are -6 and +1, since (-6) * (+1) = -6 and (-6) + (+1) = -5.

So, the factored equation is:

(u - 6)(u + 1) = 0

Step 4: Set each factor to zero and solve for 'u':

Setting u - 6 = 0: u = 6

Setting u + 1 = 0: u = -1

Step 5: Recall that we defined u as (x² - 2x - 5). Now we'll set the two expressions for u to zero and solve for 'x':

Setting x² - 2x - 5 = 6: x² - 2x - 5 - 6 = 0 x² - 2x - 11 = 0

Now we have a quadratic equation. We can either factor it or use the quadratic formula to find the solutions.

Step 6: Use the quadratic formula to find the solutions for 'x':

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = -2, and c = -11.

Plugging in the values:

x = (2 ± √((-2)² - 4 * 1 * (-11))) / 2 * 1 x = (2 ± √(4 + 44)) / 2 x = (2 ± √48) / 2 x = (2 ± 4√3) / 2

Step 7: Simplify the solutions:

x = (2 + 4√3) / 2 x = 2/2 + 4√3/2 x = 1 + 2√3

x = (2 - 4√3) / 2 x = 2/2 - 4√3/2 x = 1 - 2√3

So, the solutions for the given equation are x = 1 + 2√3 and x = 1 - 2√3.