Find the solution(s) of the quadratic equation [tex]$2 x^2+9 x+9=0$[/tex]
A. [tex]$x=3, \frac{3}{2}$[/tex]
B. [tex]$x=3$[/tex]
C. [tex]$x=-3, \frac{3}{2}$[/tex]
D. [tex]$x=-3,-\frac{3}{2}$[/tex]
Answer (1)
Factor the quadratic equation 2 x 2 + 9 x + 9 = 0 by rewriting it as 2 x 2 + 6 x + 3 x + 9 = 0 .
Factor by grouping: 2 x ( x + 3 ) + 3 ( x + 3 ) = 0 , which simplifies to ( 2 x + 3 ) ( x + 3 ) = 0 .
Set each factor to zero: 2 x + 3 = 0 and x + 3 = 0 .
Solve for x to find the solutions: x = − 2 3 and x = − 3 . The solutions are x = − 3 , x = − 2 3 .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 + 9 x + 9 = 0 . Our goal is to find the values of x that satisfy this equation.
Factoring Approach We can solve this quadratic equation either by factoring or by using the quadratic formula. Let's try factoring first. We are looking for two numbers that multiply to 2 \t \t × 9 = 18 and add up to 9 . These numbers are 6 and 3 . So we can rewrite the middle term as 6 x + 3 x .
Factoring by Grouping Now, we rewrite the equation as 2 x 2 + 6 x + 3 x + 9 = 0 . We can factor by grouping: 2 x ( x + 3 ) + 3 ( x + 3 ) = 0 . This simplifies to ( 2 x + 3 ) ( x + 3 ) = 0 .
Finding the Solutions Setting each factor equal to zero, we have 2 x + 3 = 0 or x + 3 = 0 . Solving for x in each case gives us 2 x = − 3 \t \t ⇒ x = − 2 3 and x = − 3 .
Final Answer Therefore, the solutions to the quadratic equation 2 x 2 + 9 x + 9 = 0 are x = − 3 and x = − 2 3 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling the growth of a population. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic equation. By solving the equation, you can find when the ball hits the ground.
