Which statement best describes the equation $(x+5)^2+4(x+5)+12=0$?
A. The equation is quadratic in form because it can be rewritten as a quadratic equation with $u$ substitution $u=(x+5)$.
B. The equation is quadratic in form because when it is expanded, it is a fourth-degree polynomial.
C. The equation is not quadratic in form because it cannot be solved by using the quadratic formula.
D. The equation is not quadratic in form because there is no real solution.
Answer (1)
Substitute u = x + 5 to rewrite the equation as u 2 + 4 u + 12 = 0 .
Calculate the discriminant: Δ = 4 2 − 4 ( 1 ) ( 12 ) = − 32 .
Since Δ < 0 , the equation has no real solutions.
The equation is quadratic in form and has no real solutions: The equation is quadratic in form because it can be rewritten as a quadratic equation with u substitution u = ( x + 5 ) .
Explanation
Understanding the Problem We are given the equation ( x + 5 ) 2 + 4 ( x + 5 ) + 12 = 0 and asked to determine the statement that best describes it.
Making a Substitution Let's make a substitution to simplify the equation. Let u = x + 5 . Then the equation becomes u 2 + 4 u + 12 = 0 . This is a quadratic equation in u .
Calculating the Discriminant To determine if this quadratic equation has real solutions, we can calculate the discriminant, Δ = b 2 − 4 a c , where a = 1 , b = 4 , and c = 12 .
Analyzing the Discriminant The discriminant is Δ = 4 2 − 4 ( 1 ) ( 12 ) = 16 − 48 = − 32 . Since the discriminant is negative, the quadratic equation u 2 + 4 u + 12 = 0 has no real solutions.
Interpreting the Results Since u = x + 5 , this means that the original equation ( x + 5 ) 2 + 4 ( x + 5 ) + 12 = 0 also has no real solutions. Also, since we were able to rewrite the original equation as a quadratic equation with the substitution u = x + 5 , the original equation is quadratic in form.
Conclusion Therefore, the best statement describing the equation is: The equation is quadratic in form because it can be rewritten as a quadratic equation with u substitution u = ( x + 5 ) . Also, the equation has no real solutions.
Examples
Consider a scenario where you are designing a suspension system for a vehicle. The equation ( x + 5 ) 2 + 4 ( x + 5 ) + 12 = 0 might represent the behavior of the suspension under certain conditions. If the equation has no real solutions, it indicates that the suspension system will not exhibit stable behavior under those conditions, requiring a redesign. Understanding quadratic forms and their solutions is crucial in engineering design to ensure stability and reliability.
