Solve.
$(2 y-9)(1+y)=0$
(If there is more than one solution, separate them with commas.)
Answer (1)
Set each factor to zero: 2 y − 9 = 0 or 1 + y = 0 .
Solve the first equation: 2 y = 9 , so y = 2 9 .
Solve the second equation: y = − 1 .
The solutions are 2 9 , − 1 .
Explanation
Understanding the Problem We are given the equation ( 2 y − 9 ) ( 1 + y ) = 0 . To solve this equation, we need to find the values of y that make the equation true. This equation is already factored, which makes it easier to solve.
Setting Factors to Zero The equation ( 2 y − 9 ) ( 1 + y ) = 0 is satisfied if either of the factors is equal to zero. So, we set each factor equal to zero and solve for y .
Solving the First Equation First, let's set the first factor equal to zero: 2 y − 9 = 0 To solve for y , we add 9 to both sides of the equation: 2 y = 9 Then, we divide both sides by 2: y = 2 9
Solving the Second Equation Now, let's set the second factor equal to zero: 1 + y = 0 To solve for y , we subtract 1 from both sides of the equation: y = − 1
Final Answer We have found two solutions for y : y = 2 9 and y = − 1 . Therefore, the solutions to the equation ( 2 y − 9 ) ( 1 + y ) = 0 are y = 2 9 and y = − 1 .
Examples
Understanding how to solve factored equations is crucial in many areas, such as physics and engineering, where you might need to determine when a projectile hits the ground. For example, if the height of a projectile is given by h ( t ) = ( 5 t − 20 ) ( t + 2 ) , setting h ( t ) = 0 and solving for t tells you when the projectile is at ground level. This skill is also fundamental in economics for modeling supply and demand curves, where the equilibrium point is found by solving a system of equations that often involves factored expressions.
