What is the missing coefficient?
$\left(15 x^2+11 y^2+8 x\right)-\left(7 x^2+5 y^2+2 x\right)=\square x^2+6 y^2+6 x$
A. 4
B. 8
C. 10
D. 22
Answer (1)
Simplify the left side of the equation by combining like terms: ( 15 x 2 − 7 x 2 ) + ( 11 y 2 − 5 y 2 ) + ( 8 x − 2 x ) .
Calculate the coefficient of x 2 : 15 − 7 = 8 .
Calculate the coefficient of y 2 : 11 − 5 = 6 .
Calculate the coefficient of x : 8 − 2 = 6 . The missing coefficient is 8 .
Explanation
Understanding the Problem We are given the equation ( 15 x 2 + 11 y 2 + 8 x ) − ( 7 x 2 + 5 y 2 + 2 x ) = □ x 2 + 6 y 2 + 6 x . Our goal is to find the missing coefficient of the x 2 term.
Simplifying the Equation First, let's simplify the left side of the equation by combining like terms. We have:
( 15 x 2 + 11 y 2 + 8 x ) − ( 7 x 2 + 5 y 2 + 2 x ) = ( 15 x 2 − 7 x 2 ) + ( 11 y 2 − 5 y 2 ) + ( 8 x − 2 x )
Now, let's calculate the coefficients for each term.
Calculating the Coefficients For the x 2 term, we have 15 − 7 = 8 . So, the coefficient of x 2 is 8.
For the y 2 term, we have 11 − 5 = 6 . So, the coefficient of y 2 is 6.
For the x term, we have 8 − 2 = 6 . So, the coefficient of x is 6.
Therefore, the simplified equation is:
8 x 2 + 6 y 2 + 6 x = □ x 2 + 6 y 2 + 6 x
Finding the Missing Coefficient Now, we can compare the coefficients of x 2 on both sides of the equation to find the missing coefficient. We have:
8 x 2 + 6 y 2 + 6 x = □ x 2 + 6 y 2 + 6 x
Comparing the x 2 terms, we see that the missing coefficient is 8.
Conclusion Therefore, the missing coefficient is 8.
Examples
Understanding polynomial equations and their coefficients is crucial in various fields, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the coefficients of the quadratic equation describing its path determine its initial velocity and launch angle. By correctly identifying and manipulating these coefficients, engineers can accurately predict and control the projectile's motion, ensuring it reaches its intended target. This principle extends to many other real-world applications, where understanding and manipulating polynomial equations is essential for solving complex problems.
