$\left.\frac{ \pm i \sqrt{5}}{6}\right\}$
B) $\left{\frac{-8 \pm i \sqrt{5}}{3}\right\}$
C) $\left{\frac{-4 \pm i n}{3}\right.
$
e equation. $\sqrt{m+6}+6=m$
ual the sum of the solutions. Find $S$.
B) $S=7$
C) $S=$
=13
Answer (2)
Isolate the square root: m + 6 = m − 6 .
Square both sides: m + 6 = m 2 − 12 m + 36 .
Rearrange into a quadratic: m 2 − 13 m + 30 = 0 .
Solve by factoring: ( m − 3 ) ( m − 10 ) = 0 , so m = 3 or m = 10 . Check for extraneous solutions. Only m = 10 is valid. The sum of the solutions is 10 .
Explanation
Understanding the Problem We are given the equation m + 6 + 6 = m and asked to find the sum of the solutions.
Isolating the Square Root First, we isolate the square root term by subtracting 6 from both sides: m + 6 = m − 6
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( m + 6 ) 2 = ( m − 6 ) 2 m + 6 = m 2 − 12 m + 36
Rearranging into Quadratic Form Now, we rearrange the equation into a quadratic equation by subtracting m and 6 from both sides: 0 = m 2 − 13 m + 30
Factoring the Quadratic We can solve this quadratic equation by factoring. We look for two numbers that multiply to 30 and add to -13. These numbers are -3 and -10. Thus, we can factor the quadratic as follows: ( m − 3 ) ( m − 10 ) = 0
Finding Potential Solutions This gives us two possible solutions for m : m = 3 or m = 10
Checking for Extraneous Solutions Now, we need to check for extraneous solutions by substituting these values back into the original equation m + 6 + 6 = m . For m = 3 : 3 + 6 + 6 = 9 + 6 = 3 + 6 = 9 = 3 So, m = 3 is an extraneous solution. For m = 10 : 10 + 6 + 6 = 16 + 6 = 4 + 6 = 10 So, m = 10 is a valid solution.
Finding the Sum of Solutions Therefore, the only valid solution is m = 10 . The sum of the solutions is simply 10, since there is only one valid solution.
Final Answer Thus, S = 10 .
Examples
When designing a bridge, engineers use equations involving square roots to calculate the necessary cable lengths and support structures. Solving these equations ensures the bridge's stability and safety. Similarly, in physics, determining the velocity of an object under constant acceleration often involves solving equations with square roots, helping us understand motion and predict outcomes.
We solved the equation m + 6 + 6 = m and found that the only valid solution is m = 10 . Therefore, the sum of the solutions is S = 10 .
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