Question

Корень 7x-1=3 Корень 6+5x=x Корень x+1 = x-5 Корень x2-4x-1 = 2

Answer 1

Ответ:

1.

$\frac{4}{7}$

2.

-1.5

3.

Пуста множина

4.

-1.5

Answer 2

To solve these equations involving square roots, we'll isolate the square root terms and then square both sides of the equations to eliminate the square roots. However, it is essential to check for extraneous solutions since squaring both sides can introduce additional solutions that may not satisfy the original equations. Let's proceed step by step:

1. √(7x - 1) = 3 Square both sides: (√(7x - 1))^2 = 3^2 7x - 1 = 9 Add 1 to both sides: 7x = 10 Divide by 7: x = 10/7

2. √(6 + 5x) = x Square both sides: (√(6 + 5x))^2 = x^2 6 + 5x = x^2 Move all terms to one side: x^2 - 5x - 6 = 0 Factorize: (x - 6)(x + 1) = 0 Set each factor to zero and solve for x: x - 6 = 0 --> x = 6 x + 1 = 0 --> x = -1

3. √(x + 1) = x - 5 Square both sides: (√(x + 1))^2 = (x - 5)^2 x + 1 = x^2 - 10x + 25 Move all terms to one side: x^2 - 11x + 24 = 0 Factorize: (x - 3)(x - 8) = 0 Set each factor to zero and solve for x: x - 3 = 0 --> x = 3 x - 8 = 0 --> x = 8

4. √(x^2 - 4x - 1) = 2 Square both sides: (√(x^2 - 4x - 1))^2 = 2^2 x^2 - 4x - 1 = 4 Move all terms to one side: x^2 - 4x - 5 = 0 Factorize: (x - 5)(x + 1) = 0 Set each factor to zero and solve for x: x - 5 = 0 --> x = 5 x + 1 = 0 --> x = -1

Now, let's check the solutions in the original equations to ensure they are not extraneous:

1. √(7x - 1) = 3 √(7(10/7) - 1) = 3 √(10 - 1) = 3 √(9) = 3 3 = 3 (True)

2. √(6 + 5x) = x √(6 + 5(6)) = 6 √(6 + 30) = 6 √(36) = 6 6 = 6 (True)

√(6 + 5(-1)) = -1 √(6 - 5) = -1 √(1) = -1 1 ≠ -1 (Not True, so x = -1 is extraneous)

3. √(x + 1) = x - 5 √(3 + 1) = 3 - 5 √(4) = -2 (Not True, so x = 3 is extraneous)

√(8 + 1) = 8 - 5 √(9) = 3 (True)

4. √(x^2 - 4x - 1) = 2 √((5)^2 - 4(5) - 1) = 2 √(25 - 20 - 1) = 2 √(4) = 2 2 = 2 (True)

√((-1)^2 - 4(-1) - 1) = 2 √(1 + 4 - 1) = 2 √(4) = 2 2 = 2 (True)

Thus, the valid solutions to the original equations are: x = 6, x = 5, and x = -1. The solution x = 3 is extraneous and should be discarded.