Question

Log x+2(2x^2-5x+18)=2 через одз

Answer 1

Смотри..................

Answer 2

To solve the equation $\log{x} + 2(2x^2 - 5x + 18) = 2$ using the domain (ODZ), we need to follow these steps:

Step 1: Identify the domain of the equation. Step 2: Solve the equation within the given domain.

Step 1: Domain of the equation ($ODZ$) For the equation $\log{x} + 2(2x^2 - 5x + 18) = 2$ to be valid, the argument of the logarithm $(x)$ must be positive, as the logarithm is only defined for positive values. Therefore, the domain ($ODZ$) of the equation is $x > 0$.

Step 2: Solve the equation within the given domain. Now, let's solve the equation:

$\log{x} + 2(2x^2 - 5x + 18) = 2$

First, simplify the expression by distributing the 2 on the right side:

$\log{x} + 4x^2 - 10x + 36 = 2$

Next, bring all terms to one side of the equation:

$\log{x} + 4x^2 - 10x + 36 - 2 = 0$

$\log{x} + 4x^2 - 10x + 34 = 0$

Now, let's focus on solving for $x$ within the domain $x > 0$:

Since the logarithm term is $\log{x}$, we need to use logarithmic properties to simplify the equation further. Recall the logarithmic property:

$\log_b{A} = C$ is equivalent to $A = b^C$

In our equation, $\log{x} = \log_{10}{x}$ (common logarithm).

Therefore, we can rewrite the equation as:

$10^{\log{x}} = 10^{\log_{10}{x}} = x$

Now, the equation becomes:

$x + 4x^2 - 10x + 34 = 0$

To solve this quadratic equation, we need to find the roots:

$4x^2 - 9x + 34 = 0$

The solutions for $x$ are obtained using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 4$, $b = -9$, and $c = 34$.

Now, calculate the discriminant ($D$):

$D = b^2 - 4ac = (-9)^2 - 4(4)(34) = 81 - 544 = -463$

Since the discriminant ($D$) is negative, the quadratic equation has no real solutions within the domain $x > 0$. Thus, there are no solutions to the equation $\log{x} + 2(2x^2 - 5x + 18) = 2$ in the given domain ($ODZ: x > 0$).