Squares Equal to Negatives
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 28 of 51.
In Algebra, x^2 = -9 had no solution, because no real number squares to -9. With i it has two. Start from x^2 + 9 = 0: move the 9 across to get x^2 = -9, then take the square root of both sides: x = 3i or x = -3i.
Why two? Both (3i)^2 and (-3i)^2 equal 9i^2, which is -9, just as both 3 and -3 square to 9. The square root of both sides always brings the ± with it.
The two solutions are conjugates of each other, and that is no accident: when a quadratic with real a, b and c has a solution with i, the conjugate is the other solution.
Challenge
Solve x^2 + 25 = 0. Its solutions are imaginary.
Try it yourself
Solve for x
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 Form6Complex Solutions
Squares Equal to NegativesA Shifted SquareThe Formula Goes ComplexFractions in the AnswerRecap: Complex Solutions