What are the solutions to the equation $(2 x-5)(3 x-1)=0$?
A. $x=\frac{2}{5}$ or $x=3$
B. $x=-\frac{5}{2}$ or $x=-\frac{1}{3}$
C. $x=\frac{5}{2}$ or $x=\frac{1}{3}$
D. $x=5$ or $x=1$
Answer (1)
Set each factor to zero: 2 x − 5 = 0 or 3 x − 1 = 0 .
Solve 2 x − 5 = 0 to get x = 2 5 .
Solve 3 x − 1 = 0 to get x = 3 1 .
The solutions are x = 2 5 or x = 3 1 .
Explanation
Understanding the equation The given equation is ( 2 x − 5 ) ( 3 x − 1 ) = 0 . To find the solutions, we need to set each factor equal to zero and solve for x .
Solving the first factor First, let's solve 2 x − 5 = 0 . Adding 5 to both sides gives 2 x = 5 . Dividing both sides by 2, we get x = 2 5 .
Solving the second factor Next, let's solve 3 x − 1 = 0 . Adding 1 to both sides gives 3 x = 1 . Dividing both sides by 3, we get x = 3 1 .
Final solutions Therefore, the solutions to the equation are x = 2 5 or x = 3 1 .
Examples
Understanding quadratic equations and their solutions is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. The solutions to these equations help determine the optimal dimensions and stability of the structure, ensuring it can withstand different loads and environmental conditions. Similarly, in physics, projectile motion can be described using quadratic equations, where the solutions represent the time it takes for an object to reach a certain height or distance.
