Solve the equation by using substitution:
5) [tex](t+4)^2-(t+4)-12=0[/tex]
A) {4,-3}
6) Find the distance between the
A) 244
Answer (2)
Substitute u = t + 4 into the equation.
Factor the resulting quadratic equation in u : ( u − 4 ) ( u + 3 ) = 0 .
Solve for u : u = 4 or u = − 3 .
Substitute back to find t : t = 0 or t = − 7 . The solution set is { 0 , − 7 } .
Explanation
Understanding the Problem We are given the equation ( t + 4 ) 2 − ( t + 4 ) − 12 = 0 and asked to solve it using substitution. We also need to determine if the solution set is A) {4, -3}.
Making a Substitution Let's make a substitution to simplify the equation. Let u = t + 4 .
Substituting Now, substitute u into the original equation: u 2 − u − 12 = 0
Factoring the Quadratic We can factor this quadratic equation as follows: ( u − 4 ) ( u + 3 ) = 0
Solving for u Now, we solve for u by setting each factor equal to zero:
u − 4 = 0 or u + 3 = 0
So, u = 4 or u = − 3 .
Substituting Back Next, we substitute t + 4 back in for u to solve for t :
t + 4 = 4 or t + 4 = − 3
Solving for t Solving for t in each case:
t = 4 − 4 = 0 or t = − 3 − 4 = − 7
Thus, the solution set is {0, -7}.
Comparing with the Given Option Finally, we compare our solution set {0, -7} with the given option A) {4, -3}. They are not the same.
Final Answer The solution to the equation ( t + 4 ) 2 − ( t + 4 ) − 12 = 0 is t = 0 or t = − 7 . Therefore, the solution set is { 0 , − 7 } .
Examples
Substitution is a powerful tool in mathematics that allows us to simplify complex equations by replacing a part of the expression with a single variable. This technique is widely used in various fields, such as physics and engineering, to solve differential equations and other mathematical models. For example, when analyzing the motion of a pendulum, substitution can be used to simplify the equation of motion and make it easier to solve. Similarly, in electrical circuit analysis, substitution can be used to simplify the equations governing the behavior of the circuit.
The solution to the equation (t+4)^2 - (t+4) - 12 = 0 using substitution yields the solution set {0, -7}. This does not match the given option A) {4, -3}. Hence, the solution is t = 0 or t = -7.
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