Odd Middle Terms
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 13 of 51.
When the middle number is odd, half of it is a fraction, and nothing else changes. Take x^2 + 5x + 2. Half of 5 is 5/2, and its square is 25/4. Add and subtract it: x^2 + 5x + 25/4 - 25/4 + 2.
The first three terms are (x + 5/2)^2. For the rest, write 2 as 8/4: -25/4 + 8/4 is -17/4. So the expression is (x + 5/2)^2 - 17/4, and the vertex is (-5/2, -17/4).
Keep the fractions exact. The vertex form must equal the original for every x, so rounding 25/4 to 6.3 would quietly make it a different expression.
Challenge
Write x^2 + 3x + 1 in vertex form, with exact fractions. Type each line: first the square with its borrowed number taken back, then the tidy form.
Try it yourself
Write it in vertex form
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Quadratics & Complex Numbers
2Completing the Square
Perfect Square TrinomialsThe Missing NumberCompleting the SquareIrrational AnswersWhen a Is Not 1Recap: Completing the Square5Complex Arithmetic
Adding and SubtractingMultiplyingThe ConjugateDividingA Pure Imaginary BottomRecap: Complex Arithmetic8Lines Meet Parabolas
Two CrossingsJust TouchingNo MeetingA Circle and a LineRecap: Lines and Parabolas3Vertex Form by Algebra
From Standard to VertexOdd Middle TermsA Leading NumberOpening DownRecap: Vertex Form