https://kaue.me - Mathematics - Algebra - Quadratic Equation Formula Derivation - Step By Step
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Overview
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ax^2+bx+c=0
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x^2+(bx)/a+c/a=0
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x^2+(bx)/a=-c/a
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x^2+(bx)/a+(b/(2a))^2=-c/a+(b/(2a))^2
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(x+
b/(2a))^2
=-c/a+(b/(2a))^2
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(x+
b/(2a))^2
=(b^2 - 4ac)/((2a)^2)
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x+b/(2a)=+-sqrt(b^2-4ac)/(2a)
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x=
(-b+-sqrt(b^2-4ac))/(2a)
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Overview With Main Rules
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ax^2+bx+c=0|x = y=>kx = ky|klarr1/a
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x^2+(bx)/a+c/a=0|x = y=>x + k = y + k|klarr-c/a
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x^2+(bx)/a=-c/a|x = y=>x + k = y + k|klarr
(b/(2a))^2
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x^2+(bx)/a+(b/(2a))^2=-c/a+(b/(2a))^2|
k^2 +2km + m^2
=>(k + m)^2|klarrx
m larr b/(2a)
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(x+
b/(2a))^2
=-c/a+(b/(2a))^2|k / m + p / r=>
(kr + p m) / (mr)
|klarr-cm larr a
p larr b^2
r larr (2a)^2
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(x+
b/(2a))^2
=(b^2 - 4ac)/((2a)^2)|x = y=>
sqrt(x) = sqrt(y)
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x+b/(2a)=+-sqrt(b^2-4ac)/(2a)|x = y=>x + k = y + k|klarr(-b/(2a))
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x=
(-b+-sqrt(b^2-4ac))/(2a)
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First Step
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ax^2+bx+c=0|x = y=>kx = ky
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1/a(ax^2+bx+c)=(1/a)0|
k(m + p)
=>
km + kp
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(1/a)ax^2
+(1/a)bx+(1/a)c=(1/a)0|k * 0=>0
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(1/a)ax^2
+(1/a)bx+(1/a)c=0|(k / m)p=>(kp)/m
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(1ax^2)/a
+(1bx)/a+(1c)/a=0|1 * k=>k
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(ax^2)/a+(bx)/a+c/a=0|(kp)/m=>(k / m)p
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(a/a)x^2+(bx)/a+c/a=0|k/k=>1
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1x^2+(bx)/a+c/a=0|1 * k=>k
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x^2+(bx)/a+c/a=0
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x^2+(bx)/a+c/a=0|x = y=>
x + k = y + k
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x^2+(bx)/a+c/a+(-c/a)=0+(-c/a)|k + (-k)=>0
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x^2+(bx)/a+0=0+(-c/a)|k + 0=>k
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x^2+(bx)/a=-c/a
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x^2+(bx)/a=-c/a|x = y=>x + k = y + k
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x^2+(bx)/a+(b/(2a))^2=-c/a+(b/(2a))^2
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x^2+(bx)/a+(b/(2a))^2=-c/a+(b/(2a))^2|k=>1 * k
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x^2+1((bx)/a)+(b/(2a))^2=-c/a+(b/(2a))^2|1=>k/k
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x^2+(2/2)((bx)/a)+(b/(2a))^2=-c/a+(b/(2a))^2|(k/m)(p/r)=>(kp)/(mr)
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x^2+(2bx)/(2a)+(b/(2a))^2=-c/a+(b/(2a))^2|2k=>k+k
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x^2+(bx+bx)/(2a)+(b/(2a))^2=-c/a+(b/(2a))^2|(k + m)/p=>k/p + m/p
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x^2+(bx)/(2a)+(bx)/(2a)+(b/(2a))^2=-c/a+(b/(2a))^2|(km)/p=>k(m/p)
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x*x+x(b/(2a))+(b/(2a))x+(b/(2a))(b/(2a))=-c/a+(b/(2a))^2|km + kp=>k(m + p)
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x(x+b/(2a))+b/(2a)(x+b/(2a))=-c/a+(b/(2a))^2|km + kp=>k(m + p)
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(x+b/(2a))(x+b/(2a))=-c/a+(b/(2a))^2|k * k=>k^2
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(x+b/(2a))^2=-c/a+(b/(2a))^2
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(x+b/(2a))^2=-c/a+(b/(2a))^2|(k/m)^2=>(k^2)/(m^2)
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(x+b/(2a))^2=-c/a+(b^2)/((2a)^2)|k=>1 * k
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(x+b/(2a))^2=1((-c)/a)+(b^2)/((2a)^2)|1=>k/k
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(x+b/(2a))^2=
(((2a)^2)/((2a)^2))((-c)/a)
+(b^2)/((2a)^2)|(k/m)(p/r)=>(k/r)(p/m)
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(x+b/(2a))^2=
(((2a)^2)/a)((-c)/(2a)^2)
+(b^2)/((2a)^2)|(km)^2=>k^2m^2
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(x+b/(2a))^2=
((2^2a^2)/a)((-c)/(2a)^2)
+(b^2)/((2a)^2)|k^2/k=>k
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(x+b/(2a))^2=(4a(-c))/((2a)^2)+(b^2)/((2a)^2)|km=>mk
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(x+b/(2a))^2=(-4ac)/((2a)^2)+(b^2)/((2a)^2)|k/m +p/m=>(k+p)/m
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(x+b/(2a))^2=
(-4ac+b^2)/((2a)^2)
|k+m=>m+k
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(x+b/(2a))^2=
(b^2-4ac)/((2a)^2)
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(x+b/(2a))^2=(b^2 - 4ac)/((2a)^2)|x = y=>sqrt(x) = sqrt(y)
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sqrt((x+b/(2a))^2)=sqrt((b^2-4ac)/((2a)^2))|sqrt(x^2)=>|x|
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|x+b/(2a)|=sqrt((b^2-4ac)/((2a)^2))||x| = y=>x = +-y
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x+b/(2a)=
+-sqrt((b^2-4ac)/((2a)^2))
|sqrt(k/m)=>sqrt(k)/sqrt(m)
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x+b/(2a)=
+-sqrt(b^2-4ac)/sqrt((2a)^2)
|sqrt(k^2)=>k
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x+b/(2a)=+-sqrt(b^2-4ac)/(2a)
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Last Step
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x+b/(2a)=
+-sqrt(b^2-4ac)/(2a)
|x = y=>x + k = y + k
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x+b/(2a)+(-b/(2a))=
+-sqrt(b^2-4ac)/(2a)
+(-b/(2a))|k + (-k)=>0
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x+0=
+-sqrt(b^2-4ac)/(2a)
+(-b/(2a))|k + 0=>k
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x=
+-sqrt(b^2-4ac)/(2a)
+(-b/(2a))|k + m=>m + k
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x=-b/(2a)+
+-sqrt(b^2-4ac)/(2a)
|k/m + p/m=>(k + p)/m
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x=
(-b+-sqrt(b^2-4ac))/(2a)
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