Question

Решить ³√14+x+√22-x=6

Answer 1

(+х і -х)-скоротили,тому:
3√14+√22=6
твердження хибне

Answer 2

To solve the equation ³√(14 + x) + √(22 - x) = 6, you need to isolate the variable x. Here's the step-by-step solution:

Step 1: Get rid of the radicals by raising both sides of the equation to the appropriate power to eliminate them. In this case, you'll raise both sides to the power of 3 (since we have a cube root).

(³√(14 + x) + √(22 - x))^3 = 6^3

Step 2: Expand the left-hand side (LHS) using the binomial formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

LHS = (³√(14 + x))^3 + 3 * (³√(14 + x))^2 * √(22 - x) + 3 * ³√(14 + x) * (√(22 - x))^2 + (√(22 - x))^3

Step 3: Simplify the expanded equation.

LHS = (14 + x) + 3 * ³√(14 + x) * √(22 - x) + 3 * (√(14 + x)) * (22 - x) + (22 - x)

Step 4: Combine like terms on the left-hand side.

LHS = 2x + 36 + 3 * ³√(14 + x) * √(22 - x)

Step 5: Now, we have the equation without radicals:

2x + 36 + 3 * ³√(14 + x) * √(22 - x) = 216

Step 6: Move all the terms involving x to one side of the equation:

2x + 3 * ³√(14 + x) * √(22 - x) = 216 - 36

2x + 3 * ³√(14 + x) * √(22 - x) = 180

Step 7: Now, let's isolate the cube root term:

3 * ³√(14 + x) * √(22 - x) = 180 - 2x

Step 8: Divide both sides by 3 to get the cube root term by itself:

³√(14 + x) * √(22 - x) = (180 - 2x) / 3

Step 9: Cube both sides of the equation to get rid of the cube root:

(³√(14 + x) * √(22 - x))^3 = ((180 - 2x) / 3)^3

Step 10: Expand the left-hand side.

(14 + x) * (22 - x) = ((180 - 2x) / 3)^3

Step 11: Now, simplify and solve for x.

(14 + x) * (22 - x) = (180 - 2x)^3 / 27

Expand both sides:

308 - 14x + 22x - x^2 = (180 - 2x)^3 / 27

Combine like terms:

308 + 8x - x^2 = (180 - 2x)^3 / 27

Rearrange to form a cubic equation:

x^2 - 8x + (308 - (180 - 2x)^3 / 27) = 0

Now, you have a cubic equation in terms of x. To find the solutions, you can either try to solve it by factoring or use numerical methods such as the Newton-Raphson method or graphical methods. These methods can be computationally intensive, but they will give you an approximate solution for x.