Question

1) √x^{2}-4х+4 + √x^{2}-6х+9= 1 2)√11х+3-√2-х = √9х+7 -√х-2 3)√х-5 - √10-х = 3

Answer 1

3)10-x=3    1) x^{2} -6x+9=1

-x=3-10          x^{2}-6x+9-1=0

-x=-7              x^{2}-2x-4x+9-1=0

x=7                 x^{2}-2x-4x+8=0  

                      x ×(x-2)-4x+8=0

                     x(x-2)-4(x-2)=0

                     (x-2)×(x-4)=0

                     x-2=0

                       x-4=0

                        x=2

                         x-4=0

                         x1=2

                         x2=4

Answer 2

To solve these equations, we'll follow a step-by-step process. We'll isolate the radicals on one side and move all other terms to the opposite side of the equation. Then we'll square both sides to eliminate the square roots and solve for the variable.

1. √x^2 - 4x + 4 + √x^2 - 6x + 9 = 1

First, notice that the expression under the first radical (√x^2 - 4x + 4) simplifies to (x - 2), and the expression under the second radical (√x^2 - 6x + 9) simplifies to (x - 3):

x - 2 + x - 3 = 1

Combine like terms:

2x - 5 = 1

Now, isolate the variable:

2x = 1 + 5 2x = 6

Finally, solve for x:

x = 6/2 x = 3

2. √11x + 3 - √2 - x = √9x + 7 - √x - 2

Let's handle one square root at a time:

√11x + 3 - √2 - x = √9x + 7 - √x - 2

Move the terms containing radicals to one side and the constant terms to the other side:

√11x + 3 - √9x - 7 = √x - √2 - x + √x

Combine like terms:

√11x - √9x + 3 - 7 = √x - x + √x - √2

Simplify:

√11x - 3√x - 4 = √x - x + √x - √2

Combine like terms again:

√11x - 3√x - √x + x + √x = √2 + 4

Simplify further:

√11x = 5√x + √2 + 4

Now, square both sides to eliminate the square root on the left side:

(√11x)^2 = (5√x + √2 + 4)^2

11x = 25x + 2√10x + 20√x + 2√2x + 16

Move all terms to one side:

11x - 25x - 2√10x - 20√x - 2√2x - 16 = 0

Combine like terms:

-14x - 2√10x - 20√x - 2√2x - 16 = 0

Now, this equation is not easily solvable because it has both x and square root terms. Please recheck the equation or provide more information if you believe there might be an error in the original equation.

3. √x - 5 - √10 - x = 3

Move the terms with radicals to one side and the constant term to the other side:

√x - √10 - x = 3 + 5

Combine the constants:

√x - √10 - x = 8

Now, let's square both sides to eliminate the square roots:

(√x)^2 - 2(√x)(√10) + (√10)^2 - x^2 = 8^2

x - 2√10x + 10 - x^2 = 64

Rearrange the equation:

-x^2 - 2√10x + x + 10 - 64 = 0

Simplify:

-x^2 + (1 - 2√10)x - 54 = 0

This equation is also not easily solvable using simple algebraic methods. If you think there might be an error or if there are additional constraints or information missing, please provide more details so that I can assist you further.