What constant should be added to the binomial x2 + 5x so that it becomes a perfect square trinomial?
The Correct Answer and Explanation is:
To make the binomial ( x^2 + 5x ) into a perfect square trinomial, we need to add a constant that will complete the square.
Solution:
A perfect square trinomial takes the form:
[
(a + b)^2 = a^2 + 2ab + b^2
]
In our case, ( x^2 + 5x ) is missing the last term, which is necessary to create a perfect square. To find the constant that will complete the square, we can follow these steps:
- Identify the coefficient of ( x ): Here, the coefficient of ( x ) is 5.
- Divide the coefficient of ( x ) by 2:
[
\frac{5}{2} = 2.5
] - Square the result:
[
(2.5)^2 = 6.25
]
So, the constant that we need to add to ( x^2 + 5x ) is ( 6.25 ).
Final Expression:
By adding 6.25 to ( x^2 + 5x ), we get:
[
x^2 + 5x + 6.25
]
Now, this expression can be written as a perfect square trinomial:
[
(x + 2.5)^2
]
Explanation:
Completing the square is a common method used in algebra to transform a quadratic expression into a perfect square trinomial. This approach is useful for solving quadratic equations, graphing parabolas, and integrating certain functions in calculus.
In our example, the binomial ( x^2 + 5x ) was missing a constant to make it a perfect square. The process involved finding half of the ( x )-term coefficient (5), squaring it, and then adding this square (6.25) to the binomial. The resulting trinomial, ( x^2 + 5x + 6.25 ), can then be factored as ( (x + 2.5)^2 ), which is the square of a binomial.
By adding 6.25, we completed the square, transforming the expression into an easier-to-manage form, useful for further algebraic manipulations or solving equations.