The solutions to a quadratic equation are $x=2$ and $x=7$. If $k$ is a nonzero constant, which of the following must be equal to the quadratic equation?
$(x-2)(x-7)+k=0$
$k(x-2)(x-7)=0$
$k(x+2)(x+7)=0$
$(x+2)(x+7)+k=0$
Answer (1)
A quadratic equation with roots 2 and 7 has the form a ( x − 2 ) ( x − 7 ) = 0 .
Check each option to see if it matches the required form.
Option k ( x − 2 ) ( x − 7 ) = 0 has roots 2 and 7.
The final answer is k ( x − 2 ) ( x − 7 ) = 0 .
Explanation
Understanding the Problem We are given that the solutions to a quadratic equation are x = 2 and x = 7 . We need to determine which of the given equations must be equal to the quadratic equation, given that k is a nonzero constant.
General Form of Quadratic Equation A quadratic equation with roots 2 and 7 can be written in the form a ( x − 2 ) ( x − 7 ) = 0 , where a is a nonzero constant. We need to find which of the given options matches this form.
Analyzing Each Option Let's analyze each option:
( x − 2 ) ( x − 7 ) + k = 0 : This equation can be rewritten as x 2 − 9 x + 14 + k = 0 . The roots of this equation are not necessarily 2 and 7. For example, if k = 1 , the equation becomes x 2 − 9 x + 15 = 0 , and the roots are 2 9 ± 81 − 60 = 2 9 ± 21 , which are not 2 and 7.
k ( x − 2 ) ( x − 7 ) = 0 : Since k is a nonzero constant, this equation has roots x = 2 and x = 7 . Thus, this is a possible quadratic equation.
k ( x + 2 ) ( x + 7 ) = 0 : The roots of this equation are x = − 2 and x = − 7 , which are not 2 and 7.
( x + 2 ) ( x + 7 ) + k = 0 : This equation can be rewritten as x 2 + 9 x + 14 + k = 0 . The roots of this equation are not necessarily -2 and -7. For example, if k = 1 , the equation becomes x 2 + 9 x + 15 = 0 , and the roots are 2 − 9 ± 81 − 60 = 2 − 9 ± 21 , which are not -2 and -7.
Conclusion Therefore, the only equation that must be equal to the quadratic equation is k ( x − 2 ) ( x − 7 ) = 0 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. The roots of the equation represent the points where the arch meets the ground. By manipulating the equation with a constant factor, they can adjust the scale and dimensions of the bridge while maintaining its structural integrity. This ensures the bridge is both safe and efficient.
