Study the steps shown to solve the given equation.
[tex]\begin{array}{c}
\sqrt{30-2 x}=x-3 \\
30-2 x=x^2-6 x+9 \\
0=x^2-4 x-21 \\
0=(x+3)(x-7)
\end{array}[/tex]
Based on the above work, possible solutions of the equation are [$\square$].
Answer (2)
The only valid solution to the equation 30 − 2 x = x − 3 is x = 7 . The other potential solution, x = − 3 , does not satisfy the original equation. Therefore, the answer is 7 .
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Factor the quadratic equation to find possible solutions: x = − 3 and x = 7 .
Substitute x = − 3 into the original equation: 30 − 2 ( − 3 ) = 6 = − 6 , so x = − 3 is not a solution.
Substitute x = 7 into the original equation: 30 − 2 ( 7 ) = 4 = 7 − 3 , so x = 7 is a solution.
The only valid solution is 7 .
Explanation
Analyzing the Problem We are given the equation 30 − 2 x = x − 3 and the steps to solve it. Our goal is to find the possible solutions based on the given work. The steps provided are:
30 − 2 x = x − 3
30 − 2 x = x 2 − 6 x + 9
0 = x 2 − 4 x − 21
0 = ( x + 3 ) ( x − 7 )
Finding Possible Solutions From the factored equation 0 = ( x + 3 ) ( x − 7 ) , we find two possible solutions: x = − 3 and x = 7 . However, we need to check if these solutions are valid by substituting them back into the original equation 30 − 2 x = x − 3 .
Checking x = -3 Let's check x = − 3 :
30 − 2 ( − 3 ) = 30 + 6 = 36 = 6 x − 3 = − 3 − 3 = − 6 Since 6 = − 6 , x = − 3 is not a valid solution.
Checking x = 7 Now let's check x = 7 :
30 − 2 ( 7 ) = 30 − 14 = 16 = 4 x − 3 = 7 − 3 = 4 Since 4 = 4 , x = 7 is a valid solution.
Conclusion Therefore, the only valid solution to the equation 30 − 2 x = x − 3 is x = 7 .
Examples
When designing a bridge, engineers use equations involving square roots to calculate the tension and compression forces in the supporting cables. Solving these equations ensures the bridge's stability and safety. Similarly, in physics, determining the velocity of an object in projectile motion often involves solving equations with square roots. These calculations are crucial for predicting the object's trajectory and impact point.
