Solve for x.
$(t)^2-(t+4)-12=0$
Find the distance between the points: $(-8,2)$ and $(4,-8)$
Answer (2)
Substitute u = t + 4 into the equation and factor the resulting quadratic equation in u .
Solve for u and then substitute back to find the values of t .
Use the distance formula to calculate the distance between the two points.
The solution to the equation is t = 0 or t = − 7 , and the distance between the points is 2 61 .
C
Explanation
Problem Analysis We are given the equation ( t + 4 ) 2 − ( t + 4 ) − 12 = 0 and asked to solve for t using substitution. We are also asked to find the distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) .
Substitution Let u = t + 4 . Substituting this into the equation, we get u 2 − u − 12 = 0 .
Factoring We can factor the quadratic equation in u as ( u − 4 ) ( u + 3 ) = 0 .
Solving for u Solving for u , we have u = 4 or u = − 3 .
Solving for t Substituting back t + 4 for u , we get t + 4 = 4 or t + 4 = − 3 . Solving for t , we find t = 0 or t = − 7 . Thus, the solution set for t is 0 , − 7 .
Calculating the Distance Now, we need to find the distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) . The distance formula is given by $d =
( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Substituting the coordinates of the points, we have
d = ( 4 − ( − 8 ) ) 2 + ( − 8 − 2 ) 2 = ( 12 ) 2 + ( − 10 ) 2 = 144 + 100 = 244 = 2 61
Final Answer The solution set for the equation is 0 , − 7 , which corresponds to option C. The distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) is 2 61 . None of the given options for the distance is correct. However, the number 244 is the square of the distance.
Conclusion Therefore, the solution to the equation is t = 0 or t = − 7 , and the distance between the points is 2 61 .
Examples
Understanding quadratic equations and distance calculations is crucial in various fields. For instance, in physics, projectile motion can be modeled using quadratic equations to determine the trajectory of an object. The distance formula is essential in navigation and mapping, helping to calculate distances between locations. These mathematical tools enable accurate predictions and measurements in real-world scenarios.
The solutions to the equation ( t ) 2 − ( t + 4 ) − 12 = 0 are t = 2 1 + 65 and t = 2 1 − 65 . The distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) is 2 61 .
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