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Fill in the blanks with numbers to make a true equation.

 

3x^2 + 12x - 4 - 2x^2 + 6x + 7 = ___ (x + ___ )^2 + ___

 Dec 17, 2023
 #1
avatar+222 
0

To arrive at a true equation, we need to combine the polynomial terms and complete the square for the remaining quadratic term. Let's break down the steps:

 

Combine like terms:

First, combine the x^2 terms and the constant terms:

 

(3x^2 - 2x^2) + (12x + 6x) + (-4 + 7) = ___ (x + ___ )^2 + ___

Simplify:

 

x^2 + 18x + 3 = ___ (x + ___ )^2 + ___

Complete the square for the remaining quadratic term:

 

In order to complete the square, we need to determine half of the coefficient of the x term and square it. The coefficient of the x term is 18, so half of it is 9 and squaring it gives us 81.

 

We need to add and subtract 81 to ensure that the equation remains true:

x^2 + 18x + 3 + 81 - 81 = ___ (x + ___ )^2 + ___

 

Rewrite as a squared term:

 

Now, we can rewrite the expression as a squared term:

 

(x^2 + 18x + 81) - 78 = ___ (x + ___ )^2 + ___

(x + 9)^2 - 78 = ___ (x + ___ )^2 + ___

 

Set equals to the target expression:

 

Finally, the target expression on the right side needs to be a sum of the squared term and a constant term. Since the squared term is already defined, we simply add the remaining constant term, -78, to complete the equation:

 

(x + 9)^2 - 78 = (x + ___ )^2 + (-78)

 

Therefore, the blanks should be filled with:

 

First blank: 9

Second blank: Any number, as the constant term on the right side cancels out.

 

For example, the equation:

 

(x + 9)^2 - 78 = (x + 0)^2 - 78

 

is also true. This demonstrates that the second blank does not affect the final result.

 

So, any valid answer would have the first blank filled with 9 and the second blank filled with any number.

 Dec 17, 2023

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