You have learned three methods for solving equations of the form When you can solve by using the square root property For example If the expression is factorable then you can use the Product Property of Zero ID: 737478
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Slide1
Completing the SquareSlide2
Methods of Solving Quadratic Equations
You have learned three methods for solving equations of the form
When
, you can solve by using the square root propertyFor example: If the expression is factorable, then you can use the Product Property of ZeroFor example: You can use the Quadratic Formula
Slide3
Completing the Square
You have one more method of solving to learn
First, you must learn a new technique called
completing the squareTo begin to understand this technique, you will start by performing the following multiplications and looking for a pattern Slide4
Completing the Square
The results are as follows
Do you detect a pattern?Sometimes this pattern is difficult to detect unless we choose not to simplify the multiplication of the numbers involvedHere are the results again, but without having carried out multiplication Slide5
Completing the Square
Can you see the pattern?
If you have an expression
, how would you describe the pattern (what happens to
a
when you square the binomial)?
If the terms are
x
and
a, then you might say thatYou square the first term to get Then you multiply the second term by 2 and then by x to get Finally, you square the second term to get The result is
Slide6
Completing the Square
Now that you know that
, you can tell at a glance whether a quadratic expression is a perfect square binomial
For example, is a perfect square binomial because, using the pattern of , we have that so that , and is the last termTherefore, we can write (that is, we can factor as) Slide7
Completing the Square
Example: is the expression
a perfect square binomial?
Using the pattern , we see that so that But Therefore, this is not a perfect square binomial; it cannot be factored as Slide8
Guided Practice
Tell whether the expression is a perfect square binomial. If it is, write it as a perfect square binomial.
Slide9
Guided Practice
Tell whether the expression is a perfect square binomial. If it is, write it as a perfect square binomial.
not a perfect square binomial not a perfect square binomial Slide10
Completing the Square
The technique of completing the square requires us to change a quadratic expression that is
not
a perfect square binomial, into an expression that includes a perfect square binomialUsing numbers, note that 17 is not a perfect squareBut and since , then We have taken a number that is not a perfect square, and rewritten it as an expression that includes a perfect square Slide11
Completing the Square
To do this with a quadratic expression of the form
that is not a perfect, we must first determine using the value of
b what would have to be added to make it a perfect squareFor example, you saw in the guided practice that is not a perfect square because if then , but So is a perfect square, but is not a perfect square Slide12
Completing the Square
However, we can use the inverse property for addition to fix this expression so that it contains a perfect square
We do this by adding zero, but in the form of
Since , then we can rewrite the expression asThis is a true statement since we have added zeroNow, We have written the expression so that it includes a perfect square; this is completing the square Slide13
Completing the Square
Let’s develop a consistent method for completing the square
If
, how are a, b, and c related?We saw that and that Since we will use b to determine c, then we note that and therefore This would mean that Slide14
Completing the Square Examples
Now let’s use
to work some examples
Example: complete the square for Here, so Therefore, is a perfect squareRewrite the expression by adding and subtracting 4Complete the square and combine like terms to get
Slide15
Completing the Square Examples
Now let’s use
to work some examples
Example: complete the square for Here, so Therefore, is a perfect squareRewrite the expression by adding and subtracting 64Complete the square and combine like terms to get
Slide16
Completing the Square Examples
Up to now, the examples have been chosen so that
b
is an even number, but what is b is odd?Example: complete the square for Here we have so Therefore, is a perfect squareRewrite the expression, adding and subtracting Complete the square and combine like terms to get
Slide17
Completing the Square Examples
Example: complete the square for
Here we have
so Therefore, is a perfect squareRewrite the expression, adding and subtracting Complete the square and combine like terms to get
Slide18
Guided Practice
Complete the square.
Slide19
Guided Practice
Complete the square.
Slide20
Exercise 4.7a
Handout