Questions: Solve the following expression for (x). What is(are) the value(s) of the positive root(s)? [ fracx^2(0.200-x)=12.0 ]

$Solve the following expression for (x). What is(are) the value(s) of the positive root(s)? [ fracx^2(0.200-x)=12.0 ]$

Transcript text: 2:07 Question 1 of 1 Submit Solve the following expression for $x$. What is(are) the value(s) of the positive root(s)? \[ \frac{x^{2}}{(0.200-x)}=12.0 \]

Solution

Solution Steps

To solve the given equation for $ x $, we need to first rearrange the equation to isolate $ x $. This involves multiplying both sides by $ (0.200 - x) $ to eliminate the fraction. Then, we will have a quadratic equation in the standard form $ ax^2 + bx + c = 0 $. We can use the quadratic formula to find the roots of the equation. Finally, we will filter out the positive root(s) from the solutions obtained.

Step 1: Rearranging the Equation

We start with the equation

\[ \frac{x^{2}}{(0.200 - x)} = 12.0. \]

To eliminate the fraction, we multiply both sides by $ (0.200 - x) $:

\[ x^{2} = 12.0(0.200 - x). \]

Step 2: Expanding and Rearranging

Expanding the right side gives us:

\[ x^{2} = 2.4 - 12.0x. \]

Rearranging this into standard quadratic form results in:

\[ x^{2} + 12.0x - 2.4 = 0. \]

Step 3: Solving the Quadratic Equation

Using the quadratic formula $ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} $, where $ a = 1 $, $ b = 12.0 $, and $ c = -2.4 $, we find the roots. The solutions are:

\[ x \approx -12.1968 \quad \text{and} \quad x \approx 0.1968. \]

Step 4: Identifying Positive Roots

Among the solutions, the positive root is

\[ x \approx 0.1968. \]