Completing the square rewrites a quadratic as a multiple of a perfect square plus a constant. Once a quadratic equation is in that form, the square root method solves it.
Making a perfect square
Example 9.3.1. Solve .
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X: top , bottom , sides and . So
But if we write , the X is the same (top , sides and , bottom ): the part is still the square .
Problem 9.3.2. Suppose is a constant. In terms of , what must be added to to produce a quadratic that is the square of a binomial?
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First, two squares:
Now ; compare . In general
so to get , take . (From : let , giving ; let with , giving , so .) So
Check:
Anything can be turned into a square, because adding and subtracting the same number changes nothing:
Problem 9.3.3 (MEOW!!!!!!!!!). Write in the form :
(a)
(b)
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As found above,
(a), so
Scratch along the way: ; matching the constant instead, ; and asking for , i.e. . Also ,.
(b), and . First pass:
Expanding to check: , so . Forgot the :
since ; check: . (In decimals: .)
Problem 9.3.4. Suppose and . We find .
(a) Express as the square of a binomial.
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(a). Let , so . Trying values:
and then (,,,: with ,).
Attempts: set , i.e. ; write ; or . (Also considered Vieta: ,.)
Since , and :
(Matching :;;.)
Let , so :
Solving equations by completing the square
Example 9.3.5. Solve .
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so and .
With the quadratic formula instead:
Example 9.3.6. Solve .
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X: top , sides and , bottom . Factoring out of the first part:
Example 9.3.7. Solve .
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X: top , sides and .
so and .
Example 9.3.8. Solve .
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X: top , sides and , bottom .
so and .
Example 9.3.9. Solve .
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X: top , sides and , bottom . Adding inside the bracket adds , so add outside:
Problem 9.3.10. (a)
(b)
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(a) Add :
So ,, and . With the formula:
(b)
Again: ; adding ,, so ,,,. With the formula:
Example 9.3.11. Solve .
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Factor out of the first two terms:
and ; half the coefficient is . To complete the square on we need
So
that is, or .
Problem 9.3.12. and .
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Check:
Exercises
Exercise 9.3.13. Complete the square of the following quadratics.
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(a)
(b)
(c)
(d)
since .
Exercise 9.3.14. Find such that .
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Squaring gives . Alternatively, , so
so (since ).
Exercise 9.3.15. Find all such that .
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and , so
Completing the square, , so and : either or .
If :, so ,,, so or , with (check: and ).
If :,, and .
Another route: in gives
With :, so
i.e. , together with .
So
Exercise 9.3.16. Find all :
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First manipulations: , so ;;;.
Adding all five equations:
so
Completing each square:
and , so
Since all are squared, all , but since they are added, each has to be , so