Mathematics - Algebra - Quadratic Equation Formula Derivation - Step By Step

Overview

` ax^2 `` + `` bx `` + `` c `` = `` 0 `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` + `` c/a `` = `` 0 `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` `` `` = `` -c/a `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` `` `` `` `` `` `` `` `` `` `` `
` `` (x `` + `` b/(2a))^2 `` `` = `` -c/a `` + `` (b/(2a))^2 `` `` `` `` `` `` `` `` `` `` `` `
` `` (x `` + `` b/(2a))^2 `` `` = `` `` (b^2 - 4ac)/((2a)^2) `` `` `` `` `` `` `` `` `` `` `` `` `
` `` x `` + `` b/(2a) `` `` = `` `` +-sqrt(b^2-4ac)/(2a) `` `` `` `` `` `` `` `` `` `` `` `` `
` `` x `` `` `` `` = `` `` (-b+-sqrt(b^2-4ac))/(2a) `` `` `` `` `` `` `` `` `` `` `` `` `

Overview With Main Rules

` ax^2 `` + `` bx `` + `` c `` = `` 0 `` `` `` | `` x = y `` => `` kx = ky `` | `` k `` larr `` 1/a `` `` `` `
` x^2 `` + `` (bx)/a `` + `` c/a `` = `` 0 `` `` `` | `` x = y `` => `` x + k = y + k `` | `` k `` larr `` -c/a `` `` `` `
` x^2 `` + `` (bx)/a `` `` `` = `` -c/a `` `` `` | `` x = y `` => `` x + k = y + k `` | `` k `` larr `` (b/(2a))^2 `` `` `` `
` x^2 `` + `` (bx)/a `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` k^2 +2km + m^2 `` => `` (k + m)^2 `` | `` k `` larr `` x `` m larr b/(2a) `` `` `
` `` (x `` + `` b/(2a))^2 `` `` = `` -c/a `` + `` (b/(2a))^2 `` | `` k / m + p / r `` => `` (kr + p m) / (mr) `` | `` k `` larr `` -c `` m larr a `` p larr b^2 `` r larr (2a)^2 `
` `` (x `` + `` b/(2a))^2 `` `` = `` `` (b^2 - 4ac)/((2a)^2) `` `` | `` x = y `` => `` sqrt(x) = sqrt(y) `` `` `` `` `` `` `` `
` `` x `` + `` b/(2a) `` `` = `` `` +-sqrt(b^2-4ac)/(2a) `` `` | `` x = y `` => `` x + k = y + k `` | `` k `` larr `` (-b/(2a)) `` `` `` `
` `` x `` `` `` `` = `` `` (-b+-sqrt(b^2-4ac))/(2a) `` `` `` `` `` `` `` `` `` `` `` `` `

First Step

` `` ax^2 `` + `` bx `` + `` c `` = `` 0 `` | `` x = y `` => `` kx = ky `` `` `` `` `` `` `` `` `
` 1/a `` (ax^2 `` + `` bx `` + `` c) `` = `` (1/a)0 `` | `` k(m + p) `` => `` km + kp `` `` `` `` `` `` `` `` `
` `` (1/a)ax^2 `` + `` (1/a)bx `` + `` (1/a)c `` = `` (1/a)0 `` | `` k * 0 `` => `` 0 `` `` `` `` `` `` `` `` `
` `` (1/a)ax^2 `` + `` (1/a)bx `` + `` (1/a)c `` = `` 0 `` | `` (k / m)p `` => `` (kp)/m `` `` `` `` `` `` `` `` `
` `` (1ax^2)/a `` + `` (1bx)/a `` + `` (1c)/a `` = `` 0 `` | `` 1 * k `` => `` k `` `` `` `` `` `` `` `` `
` `` (ax^2)/a `` + `` (bx)/a `` + `` c/a `` = `` 0 `` | `` (kp)/m `` => `` (k / m)p `` `` `` `` `` `` `` `` `
` `` (a/a)x^2 `` + `` (bx)/a `` + `` c/a `` = `` 0 `` | `` k/k `` => `` 1 `` `` `` `` `` `` `` `` `
` `` 1x^2 `` + `` (bx)/a `` + `` c/a `` = `` 0 `` | `` 1 * k `` => `` k `` `` `` `` `` `` `` `` `
` `` x^2 `` + `` (bx)/a `` + `` c/a `` = `` 0 `` `` `` `` `` `` `` `` `` `` `` `` `

Next Step

` x^2 `` + `` (bx)/a `` + `` c/a `` `` `` = `` 0 `` `` `` | `` x = y `` => `` x + k = y + k `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` + `` c/a `` + `` (-c/a) `` = `` 0 `` + `` (-c/a) `` | `` k + (-k) `` => `` 0 `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` + `` `` 0 `` `` = `` 0 `` + `` (-c/a) `` | `` k + 0 `` => `` k `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` `` `` `` `` = `` `` -c/a `` `` `` `` `` `` `` `` `` `` `

Next Step

` x^2 `` + `` (bx)/a `` `` `` = `` -c/a `` `` `` | `` x = y `` => `` x + k = y + k `` `` `` `` `` `` `` `
` x^2 `` + `` (bx)/a `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` `` `` `` `` `` `` `` `` `` `` `

Next Step

` `` x^2 `` + `` `` (bx)/a `` `` `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` k `` => `` 1 * k `` `` `` `
` `` x^2 `` + `` `` 1((bx)/a) `` `` `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` 1 `` => `` k/k `` `` `` `
` `` x^2 `` + `` `` (2/2)((bx)/a) `` `` `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` (k/m)(p/r) `` => `` (kp)/(mr) `` `` `` `
` `` x^2 `` + `` `` (2bx)/(2a) `` `` `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` 2k `` => `` k+k `` `` `` `
` `` x^2 `` + `` `` (bx+bx)/(2a) `` `` `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` (k + m)/p `` => `` k/p + m/p `` `` `` `
` `` x^2 `` + `` (bx)/(2a) `` + `` `` (bx)/(2a) `` + `` (b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` (km)/p `` => `` k(m/p) `` `` `` `
` `` x*x `` + `` x(b/(2a)) `` + `` `` (b/(2a))x `` + `` (b/(2a))(b/(2a)) `` = `` -c/a `` + `` (b/(2a))^2 `` | `` km + kp `` => `` k(m + p) `` `` `` `
` x `` (x `` + `` b/(2a)) `` + `` b/(2a) `` (x `` + `` b/(2a)) `` = `` -c/a `` + `` (b/(2a))^2 `` | `` km + kp `` => `` k(m + p) `` `` `` `
` (x `` `` `` `` + `` b/(2a)) `` (x `` + `` b/(2a)) `` = `` -c/a `` + `` (b/(2a))^2 `` | `` k * k `` => `` k^2 `` `` `` `
` (x `` `` `` `` + `` b/(2a))^2 `` `` `` `` = `` -c/a `` + `` (b/(2a))^2 `` `` `` `` `` `` `` `

Next Step

` (x `` + `` b/(2a))^2 `` = `` -c/a `` + `` (b/(2a))^2 `` | `` (k/m)^2 `` => `` (k^2)/(m^2) `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` -c/a `` + `` (b^2)/((2a)^2) `` | `` k `` => `` 1 * k `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` 1((-c)/a) `` + `` (b^2)/((2a)^2) `` | `` 1 `` => `` k/k `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` (((2a)^2)/((2a)^2))((-c)/a) `` + `` (b^2)/((2a)^2) `` | `` (k/m)(p/r) `` => `` (k/r)(p/m) `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` (((2a)^2)/a)((-c)/(2a)^2) `` + `` (b^2)/((2a)^2) `` | `` (km)^2 `` => `` k^2m^2 `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` ((2^2a^2)/a)((-c)/(2a)^2) `` + `` (b^2)/((2a)^2) `` | `` k^2/k `` => `` k `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` (4a(-c))/((2a)^2) `` + `` (b^2)/((2a)^2) `` | `` km `` => `` mk `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` (-4ac)/((2a)^2) `` + `` (b^2)/((2a)^2) `` | `` k/m +p/m `` => `` (k+p)/m `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` `` (-4ac+b^2)/((2a)^2) `` `` | `` k+m `` => `` m+k `` `` `` `` `` `` `` `` `` `
` (x `` + `` b/(2a))^2 `` = `` `` (b^2-4ac)/((2a)^2) `` `` `` `` `` `` `` `` `` `` `` `` `` `` `

Next Step

` (x+b/(2a))^2 `` = `` (b^2 - 4ac)/((2a)^2) `` | `` x = y `` => `` sqrt(x) = sqrt(y) `` `` `` `` `` `` `` `` `` `` `` `` `` `
` sqrt((x+b/(2a))^2) `` = `` sqrt((b^2-4ac)/((2a)^2)) `` | `` sqrt(x^2) `` => `` |x| `` `` `` `` `` `` `` `` `` `` `` `` `` `
` |x+b/(2a)| `` = `` sqrt((b^2-4ac)/((2a)^2)) `` | `` |x| = y `` => `` x = +-y `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x+b/(2a) `` = `` +-sqrt((b^2-4ac)/((2a)^2)) `` | `` sqrt(k/m) `` => `` sqrt(k)/sqrt(m) `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x+b/(2a) `` = `` +-sqrt(b^2-4ac)/sqrt((2a)^2) `` | `` sqrt(k^2) `` => `` k `` `` `` `` `` `` `` `` `` `` `` `` `` `
` x+b/(2a) `` = `` +-sqrt(b^2-4ac)/(2a) `` `` `` `` `` `` `` `` `` `` `` `` `` `` `` `` `` `

Last Step

` x `` + `` b/(2a) `` `` `` = `` +-sqrt(b^2-4ac)/(2a) `` `` `` | `` x = y `` => `` x + k = y + k `` `` `` `` `` `` `` `
` x `` + `` b/(2a) `` + `` (-b/(2a)) `` = `` +-sqrt(b^2-4ac)/(2a) `` + `` (-b/(2a)) `` | `` k + (-k) `` => `` 0 `` `` `` `` `` `` `` `
` x `` + `` `` 0 `` `` = `` +-sqrt(b^2-4ac)/(2a) `` + `` (-b/(2a)) `` | `` k + 0 `` => `` k `` `` `` `` `` `` `` `
` x `` `` `` `` `` = `` +-sqrt(b^2-4ac)/(2a) `` + `` (-b/(2a)) `` | `` k + m `` => `` m + k `` `` `` `` `` `` `` `
` x `` `` `` `` `` = `` -b/(2a) `` + `` +-sqrt(b^2-4ac)/(2a) `` | `` k/m + p/m `` => `` (k + p)/m `` `` `` `` `` `` `` `
` x `` `` `` `` `` = `` `` (-b+-sqrt(b^2-4ac))/(2a) `` `` `` `` `` `` `` `` `` `` `` `` `

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Updated on 2019 Jan 3.

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