Question

Lg (x-1) +lg (x+2)=lg 2

Answer 1

Решение во вложении.

Answer 2

To solve the equation, we can use the properties of logarithms and algebraic manipulations. The equation is:

lg(x - 1) + lg(x + 2) = lg(2)

First, we'll combine the logarithms using the product rule of logarithms:

lg[(x - 1)(x + 2)] = lg(2)

Now, since the logarithm is a one-to-one function, we can remove the logarithm from both sides of the equation:

(x - 1)(x + 2) = 2

Next, expand the left side:

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

Simplify the equation:

x^2 + x - 2 = 2

Now, move all terms to one side of the equation:

x^2 + x - 4 = 0

This is now a quadratic equation in standard form. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = 1, and c = -4.

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

x = (-1 ± √(1 + 16)) / 2

x = (-1 ± √17) / 2

So, the two solutions for x are:

1. x = (-1 + √17) / 2
2. x = (-1 - √17) / 2

These are the exact solutions to the equation. If you need numerical approximations, you can calculate the values using a calculator.