Answered

Find the real solution of the following radical equation.



$\begin{array}{c}

\sqrt{7 m+18}=m \\

m=[?]

\end{array}$

Verified Answer :

  • Identify the domain condition \(m \ge 0\) to ensure a real and valid solution.
  • Square both sides to transform the equation into the quadratic form \(m^2 - 7m - 18 = 0\).
  • Factor the quadratic as \((m - 9)(m + 2) = 0\) to find potential roots \(m = 9\) and \(m = -2\).
  • Verify the roots in the original equation to identify \(m = 9\) as the only valid solution.
  • Final answer: \(m = \boxed{9}\)

Explanation

  1. Analysis of the Equation and Domain

To solve the radical equation \(\sqrt{7m + 18} = m\), we must first identify the domain of the variable \(m\). For the square root \(\sqrt{7m + 18}\) to be defined, the expression under the radical must be non-negative, meaning \(7m + 18 \ge 0\), which simplifies to \(m \ge -\frac{18}{7}\). Furthermore, since the output of the principal square root function is always non-negative, the equation \(\sqrt{7m + 18} = m\) implies that \(m\) must also be non-negative (\(m \ge 0\)). Combining these conditions, we must have \(m \ge 0\).

  1. Solving the Quadratic Equation

To solve the equation, we square both sides to eliminate the radical:
\[(\sqrt{7m + 18})^2 = m^2\]
This leads to the quadratic equation:
\[7m + 18 = m^2\]
Rearranging all terms to one side gives:
\[m^2 - 7m - 18 = 0\]
We can solve this quadratic equation by factoring. We look for two numbers that multiply to \(-18\) and add to \(-7\). These numbers are \(-9\) and \(2\), so the quadratic factors as:
\[(m - 9)(m + 2) = 0\]
Setting each factor equal to zero, we find the potential solutions:
\[m = 9 \quad \text{or} \quad m = -2\]

  1. Verification and Conclusion

We must now verify these potential solutions in the original equation \(\sqrt{7m + 18} = m\) while also respecting the domain condition \(m \ge 0\).
Checking \(m = 9\):
\[\sqrt{7(9) + 18} = \sqrt{63 + 18} = \sqrt{81} = 9\]
Since \(9 = 9\), \(m = 9\) is a valid solution.
Checking \(m = -2\):
\[\sqrt{7(-2) + 18} = \sqrt{-14 + 18} = \sqrt{4} = 2\]
Since \(2 eq -2\), \(m = -2\) is an extraneous solution. Additionally, this contradicts our requirement that \(m \ge 0\).
Therefore, the only real solution is \(m = 9\).

Examples

Radical equations are frequently used in physics and engineering to relate variables involving time, speed, or distance. For instance, if you are calculating the time \(t\) it takes for an object to fall a certain distance \(d\), the relationship is often given by \(d = \frac{1}{2}gt^2\), which rearranges to \(t = \sqrt{\frac{2d}{g}}\). Solving equations like these ensures you can accurately determine parameters for motion analysis and structural design. Understanding the process of squaring and checking for extraneous solutions is fundamental in ensuring the accuracy of these calculations.

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