Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic. j2 + 14j +
The Correct Answer and Explanation is:
To complete the square for the quadratic expression ( j^2 + 14j + \square ), follow these steps:
Step 1: Identify the coefficient of ( j ).
The coefficient of ( j ) is 14.
Step 2: Divide the coefficient of ( j ) by 2.
[
\frac{14}{2} = 7
]
Step 3: Square the result from Step 2.
[
7^2 = 49
]
Step 4: Add this square to the expression.
The number that makes the quadratic ( j^2 + 14j + \square ) a perfect-square trinomial is 49.
Thus, the completed square is:
[
j^2 + 14j + 49
]
Step 5: Write the trinomial as a perfect square.
[
j^2 + 14j + 49 = (j + 7)^2
]
Explanation
Completing the square is a method used to rewrite a quadratic expression in the form of a perfect square trinomial. This form is helpful for solving quadratic equations, analyzing parabolas in graphing, or simplifying expressions. The key steps involve manipulating the expression to create a term that completes the square.
- Coefficient Analysis: The linear term’s coefficient, 14 in this case, determines the additional constant needed to form a perfect square.
- Division and Squaring: By dividing the coefficient of the linear term by 2 and squaring it, we ensure the result is the necessary constant. This process works because ( (a + b)^2 = a^2 + 2ab + b^2 ).
- Verification: Adding 49 to ( j^2 + 14j ) transforms it into ( (j + 7)^2 ), a perfect square trinomial.
Completing the square is foundational for solving quadratics, deriving the quadratic formula, and analyzing vertex forms of parabolas. It systematically restructures the quadratic expression, making it versatile for multiple applications in algebra and beyond.