Question

Решить уравнение √5х-6-√х-2=2

Answer 1

Вас скобки ставить вообще не учат?
Область определения: x >=2
√(5x-6) = 2 + √(x-2)
Возводим всё в квадрат
5x - 6 = (2 + √(x-2))^2 = 4 + 4√(x-2) + x - 2
Оставляем корень справа, остальное переносим налево.
4x - 8 = 4√(x-2)
x - 2 = √(x-2)
Опять возводим в квадрат
(x-2)^2 = x-2
x1 = 2
Делим всё на (x-2)
x - 2 = 1
x2 = 3

Answer 2

$\displaystyle\mathtt{\left\{{{(\sqrt{5x-6})^2=(\sqrt{x-2}+2)^2}\atop{5x-6\geq0~and~x-2\geq0}}\right\left\{{{5x-6=x-2+4\sqrt{x-2}+4}\atop{x\geq\frac{6}{5}~and~x\geq2}}\right}\\\\\mathtt{\left\{{{4x-8=4\sqrt{x-2}}\atop{x\geq2}}\right\left\{{{x-2=\sqrt{x-2}}\atop{x\geq2}}\right\left\{{{\left[\begin{array}{ccc}\mathtt{x-2=0}\\\mathtt{x-2=1}\end{array}\right}\atop{x\geq2}}\right\left\{{{\left[\begin{array}{ccc}\mathtt{x=2}\\\mathtt{x=3}\end{array}\right}\atop{x\geq2}}\right}$

ответ: $\mathtt{x=2;~3}$

Answer 3

To solve the equation √(5x - 6) - √(x - 2) = 2, follow these steps:

Step 1: Isolate one of the square root terms on one side of the equation. Step 2: Square both sides of the equation to eliminate the square roots. Step 3: Simplify the resulting equation and solve for x. Step 4: Check the solutions obtained in step 3 to ensure they are valid solutions for the original equation.

Let's proceed with the solution:

Step 1: Isolate one of the square root terms on one side of the equation. Move √(x - 2) to the right side of the equation by adding √(x - 2) to both sides:

√(5x - 6) = √(x - 2) + 2

Step 2: Square both sides of the equation to eliminate the square roots. Square the left side (√(5x - 6))^2 = 5x - 6. Square the right side (√(x - 2) + 2)^2 = (x - 2) + 4√(x - 2) + 4.

Now, the equation becomes:

5x - 6 = x - 2 + 4√(x - 2) + 4

Step 3: Simplify the resulting equation and solve for x. Move all terms involving x to one side and the constant terms to the other side:

5x - x = -2 - 4 + 6 + 4√(x - 2)

Simplify:

4x = 4 + 4√(x - 2)

Now, isolate the term with the square root:

4√(x - 2) = 4x - 4

Next, divide both sides by 4:

√(x - 2) = x - 1

Step 4: Square both sides again to eliminate the square root:

(√(x - 2))^2 = (x - 1)^2

x - 2 = x^2 - 2x + 1

Now, move all terms to one side to set the equation to zero:

x^2 - 2x - x + 1 + 2 = 0

x^2 - 3x + 3 = 0

Now, we have a quadratic equation. To solve it, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -3, and c = 3:

x = (3 ± √((-3)^2 - 4 * 1 * 3)) / 2 * 1

x = (3 ± √(9 - 12)) / 2

x = (3 ± √(-3)) / 2

Since the discriminant (b^2 - 4ac) is negative, the quadratic equation has no real solutions. The solutions involve complex numbers. Therefore, the equation √(5x - 6) - √(x - 2) = 2 has no real solutions.