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Select the value that completes the perfect square trinomial: [tex]x^2+24 x+[/tex] ______
144
288
12
24
Answer (1)
We are given an incomplete quadratic expression x 2 + 24 x + c and asked to find the value of c that makes it a perfect square trinomial.
We compare the given expression with the general form of a perfect square trinomial ( x + a ) 2 = x 2 + 2 a x + a 2 and identify that 2 a = 24 .
We solve for a and find a = 12 .
We calculate the constant term c = a 2 = 1 2 2 = 144 , which completes the perfect square trinomial: 144 .
Explanation
Understanding the Problem We are given the expression x 2 + 24 x + c and we want to find the value of c that makes this expression a perfect square trinomial. A perfect square trinomial can be written in the form ( x + a ) 2 = x 2 + 2 a x + a 2 .
Matching Coefficients Comparing the given expression x 2 + 24 x + c with the general form x 2 + 2 a x + a 2 , we can see that the coefficient of the x term in the given expression is 24. This corresponds to 2 a in the general form. Therefore, we have 2 a = 24 .
Solving for a Now, we solve for a by dividing both sides of the equation 2 a = 24 by 2: a = 2 24 = 12
Calculating the Constant Term The constant term c that completes the perfect square trinomial is a 2 . Since we found that a = 12 , we can calculate c as follows: c = a 2 = 1 2 2 = 144
Conclusion Therefore, the value that completes the perfect square trinomial is 144. The perfect square trinomial is x 2 + 24 x + 144 , which can be written as ( x + 12 ) 2 .
Examples
Perfect square trinomials are useful in many areas of mathematics, such as solving quadratic equations, completing the square, and simplifying expressions. For example, if you are building a square garden and know one side is 'x' meters and you want to increase the area by adding 24x square meters plus a constant area to make it a perfect square, you would need to add 144 square meters to make the garden a perfect square with side length 'x+12' meters. This ensures your garden is a perfect square, which can be easier to manage and aesthetically pleasing.
