Solving the Radical Equation for 'm'

The problem asks for the difference between the values of 'm' that satisfy the equation $2m = 5\sqrt{m} - 2$. We need to solve this equation first.

Transforming the Equation

Let $x = \sqrt{m}$. This substitution requires $x \ge 0$. Since $m = x^2$, we can rewrite the equation in terms of $x$. Substituting $m = x^2$ and $\sqrt{m} = x$ into the original equation yields:

$2x^2 = 5x - 2$

Solving the Quadratic Equation

Rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$:

$2x^2 - 5x + 2 = 0$

This quadratic equation can be solved by factoring:

• Find two numbers that multiply to $(2)(2) = 4$ and add to $-5$. These are $-4$ and $-1$.
• Rewrite the middle term: $2x^2 - 4x - x + 2 = 0$.
• Factor by grouping: $2x(x - 2) - 1(x - 2) = 0$.
• Factor out the common binomial factor $(x - 2)$: $(2x - 1)(x - 2) = 0$.

Setting each factor to zero gives the possible values for $x$:

$2x - 1 = 0 \implies x = \frac{1}{2}$

$x - 2 = 0 \implies x = 2$

Both solutions $x = \frac{1}{2}$ and $x = 2$ are valid because they satisfy the condition $x \ge 0$.

Finding the Values of 'm'

Convert the values of $x$ back to values of 'm' using the substitution $x = \sqrt{m}$:

• If $x = \frac{1}{2}$, then $\sqrt{m} = \frac{1}{2}$. Squaring both sides gives $m = (\frac{1}{2})^2 = \frac{1}{4}$.
• If $x = 2$, then $\sqrt{m} = 2$. Squaring both sides gives $m = 2^2 = 4$.

Both $m = \frac{1}{4}$ and $m = 4$ are solutions to the original radical equation.

Calculating the Difference

The final step is to find the difference between the two values of 'm':

Difference = $|4 - \frac{1}{4}| = |\frac{16}{4} - \frac{1}{4}| = |\frac{15}{4}| = \frac{15}{4}$.