Lesson 3 of 5
Rewrite a quadratic, locate its vertex, and solve with square roots.
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Worked example · 1 of 8
Worked example
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What number completes to a square? Check: half of 8 is 4 and its square is 16, giving .
Big ideas
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For , add to make a perfect square. In an expression, subtract the same number to preserve its value.
When the leading coefficient is not 1, factor it out of the and x terms before completing the square. Track its multiplication of any correction.
The vertex form has vertex . The vertex is a minimum for and a maximum for .
For with , use or . If there is one root; if there are no real roots. Check in the original equation.
Worked example
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Rewrite . Factor to , then obtain . The vertex is and the maximum is 11.
Step-through
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Rewrite in vertex form.
Whole picture, 4 steps
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Worked example
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Solve . Move 7: . Add 16 to both sides: . Thus or -3, giving or -7. Checks: and .
Worked example
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Solve . Then . Add 9: . The exact roots are and . Substitution using gives .
Mix-up alert
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Wrong idea: . Repair: the square adds 9, so write . For an equation, add 9 to both sides instead.
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Flashcards
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Card 1 of 8
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Square half of 10.
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What number completes ?
25.
Rewrite as a square plus a constant.
.
What is the vertex of ?
(-2,-5).
Why are there two roots when ?
Both 2 and -2 square to 4, giving or -1.
What must you do before completing ?
Factor 5 from the first two terms: 5()+1.
How many real solutions has ?
None.
Rewrite .
.
How do you verify vertex form?
Expand it and compare every coefficient.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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