Question

Lg(x+2)+lg(x-2)= lg(5x+10)

Answer 1

X>-2
x>2
(x^2-4=5x+10;
x^2-5x-14=0;
D=25+56=81
x1=7
x2=-2)
-2 не подходит, а значит ответ 7.

Answer 2

To solve the equation lg(x+2) + lg(x-2) = lg(5x+10), we need to use logarithmic properties to simplify the equation. Here's the step-by-step solution:

Step 1: Apply the logarithmic property log(a) + log(b) = log(ab).

lg((x+2)(x-2)) = lg(5x+10)

Step 2: Remove the logarithm on both sides.

(x+2)(x-2) = 5x + 10

Step 3: Expand and simplify the left side of the equation.

x^2 - 2x + 2x - 4 = 5x + 10

Step 4: Combine like terms.

x^2 - 4 = 5x + 10

Step 5: Move all terms to one side of the equation.

x^2 - 5x - 14 = 0

Step 6: Factor the quadratic equation.

(x - 7)(x + 2) = 0

Step 7: Set each factor to zero and solve for x.

x - 7 = 0 --> x = 7 x + 2 = 0 --> x = -2

Step 8: Check the solutions.

Since logarithms are only defined for positive numbers, we need to check if both solutions are valid. If they make the argument of the logarithms negative or zero, they are not valid solutions.

For x = 7: lg(7+2) + lg(7-2) = lg(5*7+10) lg(9) + lg(5) = lg(45) (logarithms of 9 and 5 are both positive)

For x = -2: lg(-2+2) + lg(-2-2) = lg(5*(-2)+10) lg(0) + lg(-4) = lg(0) (logarithm of 0 is not defined)

So, the only valid solution is x = 7.