Which quadratic equation is equivalent to $(x-4)^2-(x-4)-6=0$?
$ (u-4)^2-(u-4)-6=0$ where $u=(x-4)$
$u^2-(u-4)-6=0$ where $u=(x-4)$
$u^2-16-u-6=0$ where $u=(x-4)$
$u^2-u-6=0$ where $u=(x-4)$
Answer (2)
Substitute u = x − 4 into the equation.
Simplify the equation to get u 2 − u − 6 = 0 .
The equivalent quadratic equation is u 2 − u − 6 = 0 .
Explanation
Understanding the Problem We are given the equation ( x − 4 ) 2 − ( x − 4 ) − 6 = 0 and the substitution u = x − 4 . We want to find the equivalent quadratic equation in terms of u .
Making the Substitution Substitute u = x − 4 into the given equation ( x − 4 ) 2 − ( x − 4 ) − 6 = 0 . This gives us u 2 − u − 6 = 0 .
The Equivalent Equation Therefore, the equivalent quadratic equation is u 2 − u − 6 = 0 .
Examples
Understanding quadratic equations is crucial in many fields, such as physics and engineering. For example, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. Similarly, in physics, projectile motion can be described using quadratic equations, allowing us to predict the trajectory of a ball thrown in the air. By understanding the properties of quadratic equations, we can solve real-world problems related to optimization, trajectory prediction, and structural design.
The equivalent quadratic equation is u 2 − u − 6 = 0 where u = ( x − 4 ) . This result is derived from substituting u into the original equation. The answer option chosen is u 2 − u − 6 = 0 .
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