What are the values of $a, b$, and $c$ in the quadratic equation $0=5 x-4 x^2-2$?
A. $a=5, b=4, c=2$
B. $a=5, b=-4, c=-2$
C. $a=-4, b=5, c=-2$
D. $a=4, b=-5, c=-2$
Answer (1)
Rewrite the given quadratic equation in the standard form a x 2 + b x + c = 0 .
Compare the coefficients of the rewritten equation with the standard form.
Identify the values of a , b , and c by matching the coefficients.
The values are a = − 4 , b = 5 , and c = − 2 , so the final answer is a = − 4 , b = 5 , c = − 2 .
Explanation
Understanding the Problem We are given the quadratic equation 0 = 5 x − 4 x 2 − 2 and asked to find the values of a , b , and c when the equation is written in the standard form a x 2 + b x + c = 0 .
Rewriting the Equation To find the values of a , b , and c , we need to rewrite the given equation in the standard form. The given equation is 0 = 5 x − 4 x 2 − 2 . Rearranging the terms, we get − 4 x 2 + 5 x − 2 = 0 .
Identifying the Coefficients Now, we can compare the rewritten equation − 4 x 2 + 5 x − 2 = 0 with the standard form a x 2 + b x + c = 0 . By comparing the coefficients, we can identify the values of a , b , and c . We have:
a = − 4 b = 5 c = − 2
Final Answer Therefore, the values of a , b , and c are a = − 4 , b = 5 , and c = − 2 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the height of the projectile can be modeled by a quadratic equation. By identifying the coefficients a , b , and c , we can determine key characteristics of the trajectory, such as the maximum height and the range of the projectile. This allows engineers to design systems that accurately predict and control the motion of objects.
