Which equations are true for [tex]x=-2[/tex] and [tex]x=2[/tex]? Select two options.
[tex]x^2-4=0[/tex]
[tex]x^2=-4[/tex]
[tex]3 x^2+12=0[/tex]
[tex]4 x^2=16[/tex]
[tex]2(x-2)^2=0[/tex]
Answer (2)
Substitute x = − 2 and x = 2 into each equation.
Check if the equation holds true for both values.
x 2 − 4 = 0 is true for both values.
4 x 2 = 16 is true for both values.
The two equations that are true for both x = − 2 and x = 2 are x 2 − 4 = 0 and 4 x 2 = 16 .
Explanation
Understanding the Problem We are given 5 equations with variable x. We need to find two equations that are true for both x = -2 and x = 2.
Solution Plan Let's substitute x = − 2 and x = 2 into each of the 5 equations and check if the equation is true for both values of x.
Checking Each Equation
x 2 − 4 = 0 For x = − 2 : ( − 2 ) 2 − 4 = 4 − 4 = 0 . True. For x = 2 : ( 2 ) 2 − 4 = 4 − 4 = 0 . True. So, the equation x 2 − 4 = 0 is true for both x = − 2 and x = 2 .
x 2 = − 4 For x = − 2 : ( − 2 ) 2 = 4 e q − 4 . False. For x = 2 : ( 2 ) 2 = 4 e q − 4 . False. So, the equation x 2 = − 4 is false for both x = − 2 and x = 2 .
3 x 2 + 12 = 0 For x = − 2 : 3 ( − 2 ) 2 + 12 = 3 ( 4 ) + 12 = 12 + 12 = 24 e q 0 . False. For x = 2 : 3 ( 2 ) 2 + 12 = 3 ( 4 ) + 12 = 12 + 12 = 24 e q 0 . False. So, the equation 3 x 2 + 12 = 0 is false for both x = − 2 and x = 2 .
4 x 2 = 16 For x = − 2 : 4 ( − 2 ) 2 = 4 ( 4 ) = 16 . True. For x = 2 : 4 ( 2 ) 2 = 4 ( 4 ) = 16 . True. So, the equation 4 x 2 = 16 is true for both x = − 2 and x = 2 .
2 ( x − 2 ) 2 = 0 For x = − 2 : 2 ( − 2 − 2 ) 2 = 2 ( − 4 ) 2 = 2 ( 16 ) = 32 e q 0 . False. For x = 2 : 2 ( 2 − 2 ) 2 = 2 ( 0 ) 2 = 0 . True. So, the equation 2 ( x − 2 ) 2 = 0 is not true for both x = − 2 and x = 2 .
Final Answer The two equations that are true for both x = − 2 and x = 2 are x 2 − 4 = 0 and 4 x 2 = 16 .
Examples
Understanding quadratic equations and their solutions is crucial in many fields, such as physics and engineering. For example, when designing a bridge, engineers need to calculate the forces acting on the structure. These calculations often involve solving quadratic equations to ensure the bridge can withstand the loads. Similarly, in physics, projectile motion can be modeled using quadratic equations to determine the range and maximum height of a projectile.
The two equations that are true for both x = − 2 and x = 2 are x 2 − 4 = 0 and 4 x 2 = 16 .
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