Solve the following equation by first writing the equation in the form [tex]$a x^2=c$[/tex]: [tex]$15+c^2=96$[/tex]
A. [tex]$c= \pm 9$[/tex]
B. [tex]$c=9$[/tex]
C. [tex]$c= \pm 9.79$[/tex]
D. [tex]$c=9.79$[/tex]
Please select the best answer from the choices provided
Answer (2)
The solutions to the equation 15 + c 2 = 96 are c = ± 9 . This is determined by isolating c 2 and taking the square root. Therefore, the correct answer is option A ( c = ± 9 ).
;
Subtract 15 from both sides: c 2 = 96 − 15 .
Simplify: c 2 = 81 .
Take the square root of both sides: c = ± 81 .
Simplify to find the solutions: c = ± 9 .
Explanation
Problem Analysis We are given the equation 15 + c 2 = 96 and asked to solve for c . The goal is to isolate c 2 and then take the square root to find the possible values of c .
Isolating c^2 First, we subtract 15 from both sides of the equation to isolate the c 2 term: c 2 = 96 − 15
Simplifying the Equation Next, we simplify the right side of the equation: c 2 = 81
Taking the Square Root Now, we take the square root of both sides of the equation to solve for c :
c = ± s q r t 81
Simplifying the Result Finally, we simplify the square root: c = p m 9
Final Answer Therefore, the solutions for c are c = 9 and c = − 9 . This corresponds to option A.
Examples
Understanding how to solve equations like this is useful in many real-world situations. For example, if you are designing a square garden and know the area you want the garden to cover, you can use this type of equation to determine the length of each side. If the area of the garden is 81 square feet, then the side length, c , would satisfy c 2 = 81 . Solving this equation tells you that each side of the garden should be 9 feet long (we take the positive root since length cannot be negative).
