What are the solutions of the equation [tex]$y=-2 x^2+9 x-4$[/tex] shown in the graph below?
A. [tex]$x=-\frac{1}{2}, 4$[/tex]
B. [tex]$x=-4,4$[/tex]
C. [tex]$x=\frac{1}{3}, 4$[/tex]
D. [tex]$x=\frac{1}{2}, 4$[/tex]
Answer (2)
The solutions to the equation y = − 2 x 2 + 9 x − 4 are x = 2 1 and x = 4 . The correct multiple choice option is D. x = 2 1 , 4 .
;
Set the quadratic equation to zero: − 2 x 2 + 9 x − 4 = 0 .
Factor the quadratic equation: ( 2 x − 1 ) ( x − 4 ) = 0 .
Solve for x by setting each factor to zero: 2 x − 1 = 0 and x − 4 = 0 .
The solutions are x = 2 1 and x = 4 , so the answer is x = 2 1 , 4 .
Explanation
Understanding the Problem We are given the quadratic equation y = − 2 x 2 + 9 x − 4 and asked to find its solutions, which are the x -intercepts. The x -intercepts occur where y = 0 .
Setting up the Equation To find the solutions, we set y = 0 and solve the equation − 2 x 2 + 9 x − 4 = 0 .
Simplifying the Equation We can multiply the equation by − 1 to make the leading coefficient positive: 2 x 2 − 9 x + 4 = 0 .
Factoring the Quadratic Now, we can factor the quadratic expression. We look for two numbers that multiply to 2 × 4 = 8 and add up to − 9 . These numbers are − 1 and − 8 . So we can rewrite the middle term as − 8 x − x :
2 x 2 − 8 x − x + 4 = 0
Factoring by Grouping Next, we factor by grouping:
2 x ( x − 4 ) − 1 ( x − 4 ) = 0
( 2 x − 1 ) ( x − 4 ) = 0
Solving for x Now, we set each factor equal to zero and solve for x :
2 x − 1 = 0 or x − 4 = 0
Finding the Solutions Solving 2 x − 1 = 0 , we get 2 x = 1 , so x = 2 1 .
Solving x − 4 = 0 , we get x = 4 .
Stating the Solutions Therefore, the solutions are x = 2 1 and x = 4 .
Selecting the Correct Option Comparing our solutions with the given options, we see that the correct answer is x = 2 1 , 4 .
Examples
Understanding quadratic equations is crucial in many real-world applications. For instance, engineers use them to design parabolic arches in bridges, ensuring structural stability. Similarly, physicists use quadratic equations to model projectile motion, predicting the trajectory of objects like a ball thrown in the air. By finding the roots of these equations, we can determine key points such as the maximum height or the landing point of the projectile. This knowledge helps in optimizing designs and predicting outcomes in various scientific and engineering fields.
