Which value is a solution of [tex]\frac{7}{4} x^2-2=-05 x+4[/tex]?

A. [tex]x=-2[/tex]
B. [tex]x=-1[/tex]
C. [tex]x=1[/tex]
D. [tex]x=5[/tex]

Substitute each given value into the equation.
Check if the left-hand side (LHS) equals the right-hand side (RHS).
If LHS = RHS, then that value is a solution.
The solution is x = − 2 ​ .

Explanation

Understanding the problem We are given the quadratic equation 4 7 ​ x 2 − 2 = − 0.5 x + 4 . We need to check which of the given values x = − 2 , − 1 , 1 , 5 is a solution to the equation. A value is a solution if, when substituted into the equation, the left-hand side (LHS) equals the right-hand side (RHS).

Testing x = -2 Let's test x = − 2 :
LHS: 4 7 ​ ( − 2 ) 2 − 2 = 4 7 ​ ( 4 ) − 2 = 7 − 2 = 5 RHS: − 0.5 ( − 2 ) + 4 = 1 + 4 = 5 Since LHS = RHS, x = − 2 is a solution.

Testing x = -1 Let's test x = − 1 :
LHS: 4 7 ​ ( − 1 ) 2 − 2 = 4 7 ​ − 2 = 4 7 ​ − 4 8 ​ = − 4 1 ​ RHS: − 0.5 ( − 1 ) + 4 = 0.5 + 4 = 4.5 = 2 9 ​ Since LHS  = RHS, x = − 1 is not a solution.

Testing x = 1 Let's test x = 1 :
LHS: 4 7 ​ ( 1 ) 2 − 2 = 4 7 ​ − 2 = 4 7 ​ − 4 8 ​ = − 4 1 ​ RHS: − 0.5 ( 1 ) + 4 = − 0.5 + 4 = 3.5 = 2 7 ​ Since LHS  = RHS, x = 1 is not a solution.

Testing x = 5 Let's test x = 5 :
LHS: 4 7 ​ ( 5 ) 2 − 2 = 4 7 ​ ( 25 ) − 2 = 4 175 ​ − 2 = 4 175 ​ − 4 8 ​ = 4 167 ​ RHS: − 0.5 ( 5 ) + 4 = − 2.5 + 4 = 1.5 = 2 3 ​ = 4 6 ​ Since LHS  = RHS, x = 5 is not a solution.

Conclusion Therefore, the only solution among the given values is x = − 2 .


Examples
In physics, when analyzing projectile motion, you might encounter quadratic equations that describe the trajectory of a ball thrown in the air. Solving these equations helps determine the launch angle needed to hit a target or the time it takes for the ball to reach a certain height. Similarly, in engineering, quadratic equations are used to design suspension bridges, ensuring they can withstand specific loads and stresses. By finding the solutions to these equations, engineers can optimize the bridge's structure for maximum stability and safety.

Answered by GinnyAnswer | 2025-07-06

After substituting the given values into the equation 4 7 ​ x 2 − 2 = − 0.5 x + 4 , only x = − 2 resulted in both sides being equal. Therefore, the correct answer is x = − 2 ​ .
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Answered by Anonymous | 2025-08-24