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Completing the Square
To complete the square for the expression x^{2} + bx , add
, which is the square of
half the coefficient of x. Consequently,
When solving quadratic equations by completing the square,
you must add to both
sides to maintain equality.
Completing the Square: Leading Coefficient is 1
Let’s solve the equation x^{2} − 6x + 2 = 0 by completing the square.
x^{2} + 6x + 2 = 0  Original Equation 
x^{2} + 6x = −2  Subtract 2 from both sides 
Divide the 6 by 2, square it, and then add to both sides  
x^{2} + 6x + 9 = 7  Simplify 
(x + 3)^{2} = 7  Perfect square trinomial 
Extract square roots  
Solutions 
Completing the Square: Leading Coefficient is Not 1
Let’s solve the equation 3x^{2} − 4x − 5 = 0 by completing the square.
If the leading coefficient of a quadratic equation is not 1, you should divide
both sides of
the equation by this coefficient before completing the square.
3x^{2} − 4x − 5 = 0  Original equation 
3x^{2} − 4x = 5  Add 5 to both sides 
Divide both sides by 3  
Divide by 2, square it, and then add to both sides  
Perfect square trinomial  
Extract square roots  
Solutions 
Using a graphing calculator , you can see that the two
solutions are approximately
2.11963 and –0.78630, which agree with the two graphical solutions shown below.
Completing the Square: One Term is Not Present
Let’s solve the equation 4x^{2} − 7x = 0 by completing the square.
As you can see, we have no constant but we will treat the problem the same as if
there
was a constant present . We skip the step of moving the constant over to the
other side of
the equation and continue on from there.
4x^{2} − 7x = 0  Original equation 
Divide both sides by 4  
Divide by 2, square it, and then add to both sides  
Perfect square trinomial  
Extract square roots  
Solutions 
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