Select the value that completes the perfect square trinomial: [tex]x^2-14 x+[/tex] ______
A) 28
B) -49
C) -28
D) 49
Answer (2)
Recognize the form of a perfect square trinomial: ( x − a ) 2 = x 2 − 2 a x + a 2 .
Find a such that − 2 a = − 14 , which gives a = 7 .
Calculate c = a 2 = 7 2 = 49 .
The value that completes the perfect square trinomial is 49 .
Explanation
Understanding the Problem We are given an incomplete quadratic expression: x 2 − 14 x + c , where c is a constant. Our goal is to find the value of c that makes the expression a perfect square trinomial.
Recognizing the Perfect Square Form A perfect square trinomial can be written in the form ( x + a ) 2 = x 2 + 2 a x + a 2 or ( x − a ) 2 = x 2 − 2 a x + a 2 . In our case, we have x 2 − 14 x + c , which resembles the second form.
Solving for a We want to find a value a such that − 2 a = − 14 . Solving for a , we get: − 2 a = − 14 ⟹ a = − 2 − 14 = 7. So, a = 7 .
Calculating c Now we need to find c , which corresponds to a 2 in the perfect square trinomial. Since a = 7 , we have: c = a 2 = 7 2 = 49.
Conclusion Therefore, the perfect square trinomial is x 2 − 14 x + 49 , which can be factored as ( x − 7 ) 2 . The value that completes the perfect square trinomial is 49.
Examples
Perfect square trinomials are useful in many areas, such as completing the square to solve quadratic equations, simplifying expressions in calculus, and in physics when dealing with oscillatory motion. For example, consider a simple harmonic oscillator where the equation of motion involves a quadratic form. By completing the square, we can easily find the equilibrium point and analyze the system's behavior. In engineering, perfect square trinomials can help in designing systems that minimize energy loss or maximize efficiency.
The value that completes the perfect square trinomial x 2 − 14 x + c is 49 , making the correct answer D ) 49 .
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