Solving Quadratic Equations

Lesson 3: The nature of the roots

From previous discussions, we know that an equation of the form y=a x 2 +bx+c MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0de9zqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadggacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyaiaadIhacqGHRaWkcaWGJbaaaa@3F49@  has two roots, but the roots are not always distinct. Take, for example, the equation

y = x 2 +4x+4 =( x+2 )( x+2 ). MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0de9zqqrpepC0xbbL8F4rqqrFfpi0xe9LqFHe9Lqpepeea0xd9q8as0=LqLs=Jirpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaaqaaiaadMhaaeaacqGH9aqpcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiaadIhacqGHRaWkcaaI0aaabaaabaGaeyypa0ZaaeWaaeaacaWG4bGaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaabaGaamiEaiabgUcaRiaaikdaaiaawIcacaGLPaaacaGGUaaaaaaa@4A9B@