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SOLVING QUADRATIC EQUATION : Solving by Quadratic Formula

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Introduction
Apart from factoring method, quadratic equation can be solved by quadratic formula and given as below,

x= −b± b 2 −4ac 2a MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8XjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcbaGaamiEaiabg2da9maalaaabaGaeyOeI0IaamOyaiabgglaXoaakaaabaGaamOyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaisdacaWGHbGaam4yaaWcbeaaaOqaaiaaikdacaWGHbaaaaaa@458B@

How to - Solving Quadratics Equation : Solving by Quadratic Formula?

To solve a quadratic equation by quadratic formula.
  1.  Transform the equation into standard form. 
  2. Obtain the values of a, b and c from given equation. 
  3. Substitute the three values into quadratic formula. 
  4. Calculate the two values of x 
Example : Solve the following by using quadratic formula.
a) x 2 +6x+5=0 a=1,  b=6,  c=5 x= −b± b 2 −4ac 2a x= −( 6 )± ( 6 ) 2 −4( 1 )( 5 ) 2( 1 ) x= −6± 16 2 x= −6±4 2 so,x= −6+4 2 ,  x= −6−4 2 x=−1,  x=−5 ¯ ¯ MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8XjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@A26A@

b)  x 2 +10x+9=0 a=1,  b=10,  c=9 x= −b± b 2 −4ac 2a = −( 10 )± ( 10 ) 2 −4( 1 )( 9 ) 2( 1 ) = −10± 64 2 = −10±8 2 x= −10+8 2 ,  x= −10−8 2     x=−1,  x=−9 ¯ ¯ MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8XjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@AAD0@

c) x( x+3 )=40 x 2 +3x=40 x 2 +3x−40=0 x= −b± b 2 −4ac 2a x= −( 3 )± ( 3 ) 2 −4( 1 )( −40 ) 2( 1 ) x= −3± 169 2 x= −3±13 2 x= −3+13 2 ,  x= −3−13 2 x=5,  x=−8 ¯ ¯ MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8XjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@A456@

Here are three examples: Bear in mind that positive and negative sign of the roots play important role in this method.

Try it yourself - Solving Quadratic Equation : Solving quadratic equation by using Quadratic formula
a)  x 2 −7x=−10 b) 2 x 2 =3x c)x( x−8 )=2x−25 } ans-x=2,5 ans-x=0, 3 2 ans-x=5 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8XjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcbaWaaiGaaqaabeqaaiaadggacaGGPaGaaGPaVlaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI3aGaamiEaiabg2da9iabgkHiTiaaigdacaaIWaaabaGaamOyaiaacMcacaaMc8UaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIZaGaamiEaaqaaiaadogacaGGPaGaamiEamaabmaabaGaamiEaiabgkHiTiaaiIdaaiaawIcacaGLPaaacqGH9aqpcaaIYaGaamiEaiabgkHiTiaaikdacaaI1aaaaiaaw2haauaabeqadeaaaeaaruavHH2BTfgaiyaajugWaabbaaaaaG+acXwDLbWdbiaa=fgacaWFUbGaa83Caiaa=1capaGaamiEaiabg2da9iaaikdacaGGSaGaaGynaaGcbaqcLbmapeGaa8xyaiaa=5gacaWFZbGaa8xla8aacaWG4bGaeyypa0JaaGimaiaacYcalmaalaaabaGaaG4maaqaaiaaikdaaaaakeaajugWa8qacaWFHbGaa8NBaiaa=nhacaWFTaWdaiaadIhacqGH9aqpcaaI1aaaaaaa@78B4@

More Exercise

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