Perfect Square Trinomials
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 6 of 51.
Square a bracket and a pattern appears: (x + 5)^2 = x^2 + 10x + 25. The middle number, 10, is twice 5, and the last number, 25, is 5 squared. Every square of x + p comes out as x^2 + 2px + p^2.
A three-term expression of that shape is a perfect square trinomial, and you can run the pattern backwards. In x^2 + 6x + 9, half of 6 is 3, and 3 squared is 9, so it is (x + 3)^2. A minus in the middle gives a minus in the bracket: x^2 - 8x + 16 is (x - 4)^2, because -4 times -4 is +16.
The test is quick: halve the middle number, square it, and compare with the last number. x^2 + 6x + 8 fails, because half of 6 squared is 9, not 8. With real numbers, the last number of a perfect square is never negative.
Challenge
Write x^2 - 14x + 49 as the square of a bracket. Run the test first: halve the middle number and square it. Then type the square as the next line.
Try it yourself
Factor the expression
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 Parabolas