Factoring a Pair of Planes into Linear Equations

1. Once a General Quadratic Equation in 3 Variables has been identified as representing a Pair of Planes, it can be Factored into it's 2 Constituent Real or Complex Linear Equations.
2. Let consider a General Quadratic Equation in 3 Variables representing a Pair of Planes given as follows
   
   \(Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + K=0\)   ...(1)
   
   which has to be Factored into it's corresponding 2 Linear Equations given as follows
   
   \(A_1x + B_1y + C_1z + D_1= 0\)   ...(2)
   
   \(A_2x + B_2y + C_2z + D_2= 0\)   ...(3)
   
   Now, the Coefficients and the Constant of the equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations (2) and (3) above as follows
   
   \(A= A_1A_2\)   ...(4)
   
   \(B= B_1B_2\)   ...(5)
   
   \(C= C_1C_2\)   ...(6)
   
   \(D= A_1B_2 + A_2B_1\)   ...(7)
   
   \(E= A_1C_2 + A_2C_1\)   ...(8)
   
   \(F= B_1C_2 + B_2C_1\)   ...(9)
   
   \(G= A_1D_2 + A_2D_1\)   ...(10)
   
   \(H= B_1D_2 + B_2D_1\)   ...(11)
   
   \(I= C_1D_2 + C_2D_1\)   ...(12)
   
   \(K= D_1D_2\)   ...(13)
   
   The following gives the steps of Finding the Values of the Coefficients and Constants \(A_1, A_2, B_1, B_2, C_1, C_2, D_1\) and \(D_2\) of the Linear Equations (2) and (3) above
   1. Normalize the General Quadratic Equation in 3 Variables given in equation (1).
   2. If the Coefficient of \(x^2\) (\(A\)) in the equation (1) is Not Zero, then the Coefficient of \(x^2\) (\(A\)) in the Normalized version of equation (1) becomes \(1\). Since \(A=A_1A_2\), the Values of the Coefficients \(A_1\) and \(A_2\) can be set as \(1\) as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A = 1\)   ...(14)
      
      \(B= B_1B_2\)   ...(15)
      
      \(C= C_1C_2\)   ...(16)
      
      \(D= B_1 + B_2\)   ...(17)
      
      \(E= C_1 + C_2\)   ...(18)
      
      \(F= B_1C_2 + B_2C_1\)   ...(19)
      
      \(G= D_1 +D_2\)   ...(20)
      
      \(H= B_1D_2 + B_2D_1\)   ...(21)
      
      \(I= C_1D_2 + C_2D_1\)   ...(22)
      
      \(K= D_1D_2\)   ...(23)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient \(G\) and Constant \(K\) in equations (20) and (23) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Gx + K=0\)   ...(24)
      
      \(\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0\)   ...(25)
      
      The Values of Constants \(D_1\) and \(D_2\) can be found out Finding the Roots of Quadratic Equation (24) above.
      
      Now, if Values of either \(D_1\) or \(D_2\) or Both are Not Zero then the Values of Coefficients \(B_1\) and \(B_2\) can be found out by Solving the System of Linear Equations given by equations (7) and (11) above as follows
      
      \(A_1B_2 + A_2B_1=D\)   ...(from equation 7)
      
      \(B_1D_2 + B_2D_1=H\)   ...(from equation 11)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}D \\ H\end{bmatrix}\)   ...(26)
      
      \(\Rightarrow \begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}D \\ H\end{bmatrix}\)   ...(27)
      
      Also, the Values of Coefficients \(C_1\) and \(C_2\) can be found out by Solving the System of Linear Equations given by equations (8) and (12) above as follows
      
      \(A_1C_2 + A_2C_1=E\)   ...(from equation 8)
      
      \(C_1D_2 + C_2D_1=I\)   ...(from equation 12)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}E \\ I\end{bmatrix}\)   ...(28)
      
      \(\Rightarrow \begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}E \\ I\end{bmatrix}\)   ...(29)
      
      Otherwise, if Values of Both \(D_1\) and \(D_2\) are Zero, a Quadratic Equation can be formed by using the Values of Coefficients \(D\) and \(B\) in equations (17) and (15) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Dx + B=0\)   ...(30)
      
      \(\Rightarrow x^2 - (B_1 + B_2)x + B_1B_2=0\)   ...(31)
      
      The Values of Coefficients \(B_1\) and \(B_2\) can be found out Finding the Roots of Quadratic Equation (30) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients \(E\) and \(C\) in equations (18) and (16) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Ex + C=0\)   ...(32)
      
      \(\Rightarrow x^2 - (C_1 + C_2)x + C_1C_2=0\)   ...(33)
      
      The Values of Coefficients \(C_1\) and \(C_2\) can be found out Finding the Roots of Quadratic Equation (32) above.
      
      The values of \(B_1, C_1\) and \(B_2, C_2\) can be matched and verified by evaluating equation (19) above.
   3. If the Coefficient of \(x^2\) (\(A\)) in the equation (1) is Zero and Coefficient of \(y^2 \) (\(B\)) in the equation (1) is Not Zero, then the Coefficient of \(y^2\) (\(B\)) in the Normalized version of equation (1) becomes \(1\). Since \(B=B_1B_2\) , the Values of the Coefficients \(B_1\) and \(B_2\) can be set as \(1\) as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= A_1A_2\)   ...(34)
      
      \(B= 1\)   ...(35)
      
      \(C= C_1C_2\)   ...(36)
      
      \(D= A_1 + A_2\)   ...(37)
      
      \(E= A_1C_2 + A_2C_1\)   ...(38)
      
      \(F= C_1 + C_2\)   ...(39)
      
      \(G= A_1D_2 + A_2D_1\)   ...(40)
      
      \(H= D_1 + D_2\)   ...(41)
      
      \(I= C_1D_2 + C_2D_1\)   ...(42)
      
      \(K= D_1D_2\)   ...(43)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient \(H\) and Constant \(K\) in equations (41) and (43) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Hx + K=0\)   ...(44)
      
      \(\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0\)   ...(45)
      
      The Values of Constants \(D_1\) and \(D_2\) can be found out Finding the Roots of Quadratic Equation (44) above.
      
      Now, if Values of either \(D_1\) or \(D_2\) or Both are Not Zero then the Values of Coefficients \(A_1\) and \(A_2\) can be found out by Solving the System of Linear Equations given by equations (7) and (10) above as follows
      
      \(A_1B_2 + A_2B_1=D\)   ...(from equation 7)
      
      \(A_1D_2 + A_2D_1=G\)   ...(from equation 10)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}D \\ G\end{bmatrix}\)   ...(46)
      
      \(\Rightarrow \begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}D \\ G\end{bmatrix}\)   ...(47)
      
      Also, the Values of Coefficients \(C_1\) and \(C_2\) can be found out by Solving the System of Linear Equations given by equations (9) and (12) above as follows
      
      \(B_1C_2 + B_2C_1=F\)   ...(from equation 9)
      
      \(C_1D_2 + C_2D_1=I\)   ...(from equation 12)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}F \\ I\end{bmatrix}\)   ...(48)
      
      \(\Rightarrow \begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}F \\ I\end{bmatrix}\)   ...(49)
      
      Otherwise, if Values of Both \(D_1\) and \(D_2\) are Zero, a Quadratic Equation can be formed by using the Values of Coefficients \(D\) and \(A\) in equations (37) and (34) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Dx + A=0\)   ...(50)
      
      \(\Rightarrow x^2 - (A_1 + A_2)x + A_1A_2=0\)   ...(51)
      
      The Values of Coefficients \(A_1\) and \(A_2\) can be found out Finding the Roots of Quadratic Equation (50) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients \(F\) and \(C\) in equations (39) and (36) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Fx + C=0\)   ...(52)
      
      \(\Rightarrow x^2 - (C_1 + C_2)x + C_1C_2=0\)   ...(53)
      
      The Values of Coefficients \(C_1\) and \(C_2\) can be found out Finding the Roots of Quadratic Equation (52) above.
      
      The values of \(A_1, C_1\) and \(A_2, C_2\) can be matched and verified by evaluating equation (38) above.
   4. If the Coefficients of Both \(x^2 \) (\(A\)) and \(y^2 \) (\(B\)) in the equation (1) are Zero and Coefficient of \(z^2\) (\(C\)) in the equation (1) is Not Zero, then the Coefficient of \(z^2\) (\(C\)) in the Normalized version of equation (1) becomes \(1\). Since \(C=C_1C_2\) , the Values of the Coefficients \(C_1\) and \(C_2\) can be set as \(1\) as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= A_1A_2\)   ...(54)
      
      \(B= B_1B_2\)   ...(55)
      
      \(C= 1\)   ...(56)
      
      \(D= A_1B_2 + A_2B_1\)   ...(57)
      
      \(E= A_1 + A_2\)   ...(58)
      
      \(F= B_1 + B_2\)   ...(59)
      
      \(G= A_1D_2 + A_2D_1\)   ...(60)
      
      \(H= B_1D_2 + B_2D_1\)   ...(61)
      
      \(I= D_1 + D_2\)   ...(62)
      
      \(K= D_1D_2\)   ...(63)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient \(I\) and Constant \(K\) in equations (41) and (43) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Ix + K=0\)   ...(64)
      
      \(\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0\)   ...(65)
      
      The Values of Constants \(D_1\) and \(D_2\) can be found out Finding the Roots of Quadratic Equation (64) above.
      
      Now, if Values of either \(D_1\) or \(D_2\) or Both are Not Zero then the Values of Coefficients \(A_1\) and \(A_2\) can be found out by Solving the System of Linear Equations given by equations (8) and (10) above as follows
      
      \(A_1C_2 + A_2C_1=E\)   ...(from equation 8)
      
      \(A_1D_2 + A_2D_1=G\)   ...(from equation 10)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}E \\ G\end{bmatrix}\)   ...(66)
      
      \(\Rightarrow \begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}E \\ G\end{bmatrix}\)   ...(67)
      
      Also, the Values of Coefficients \(B_1\) and \(B_2\) can be found out by Solving the System of Linear Equations given by equations (9) and (11) above as follows
      
      \(B_1C_2 + B_2C_1=F\)   ...(from equation 9)
      
      \(B_1D_2 + B_2D_1=H\)   ...(from equation 11)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      \(\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}F \\ H\end{bmatrix}\)   ...(68)
      
      \(\Rightarrow \begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}F \\ H\end{bmatrix}\)   ...(69)
      
      Otherwise, if Values of Both \(D_1\) and \(D_2\) are Zero, a Quadratic Equation can be formed by using the Values of Coefficients \(E\) and \(A\) in equations (58) and (54) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Ex + A=0\)   ...(70)
      
      \(\Rightarrow x^2 - (A_1 + A_2)x + A_1A_2=0\)   ...(71)
      
      The Values of Coefficients \(A_1\) and \(A_2\) can be found out Finding the Roots of Quadratic Equation (70) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients \(F\) and \(B\) in equations (59) and (55) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      \(x^2 - Fx + B=0\)   ...(72)
      
      \(\Rightarrow x^2 - (B_1 + B_2)x + B_1B_2=0\)   ...(73)
      
      The Values of Coefficients \(B_1\) and \(B_2\) can be found out Finding the Roots of Quadratic Equation (72) above.
      
      The values of \(A_1, B_1\) and \(A_2, B_2\) can be matched and verified by evaluating equation (57) above.
   5. If the Coefficients of Both \(x^2 \) (\(A\)), \(y^2 \) (\(B\)) and \(z^2 \) (\(C\)) in the equation (1) are Zero and Coefficient of \(xy\) (\(D\)) in the equation (1) is Not Zero, then the Coefficient of \(xy\) (\(D\)) in the Normalized version of equation (1) becomes \(1\). Under such condition, one of the following 2 things can be done
      
      Either we can set \(A_1=B_2=1\) and \(A_2=B_1=0\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(74)
      
      \(B= 0\)   ...(75)
      
      \(C= 0\)   ...(76)
      
      \(D= 1\)   ...(77)
      
      \(E= C_2\)   ...(78)
      
      \(F= C_1\)   ...(79)
      
      \(G= D_2\)   ...(80)
      
      \(H= D_1\)   ...(81)
      
      \(I= C_1D_2+C_2D_1\)   ...(82)
      
      \(K= D_1D_2\)   ...(83)
      
      Under such condition, the Value of \(C_1=F\), \(C_2=E\), \(D_1=H\) and \(D_2=G\) as given by equations (79), (78), (81) and (80) above respectively.
      
      Otherwize, we can set \(A_1=B_2=0\) and \(A_2=B_1=1\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(74)
      
      \(B= 0\)   ...(75)
      
      \(C= 0\)   ...(76)
      
      \(D= 1\)   ...(77)
      
      \(E= C_1\)   ...(78)
      
      \(F= C_2\)   ...(79)
      
      \(G= D_1\)   ...(80)
      
      \(H= D_2\)   ...(81)
      
      \(I= C_1D_2+C_2_D_1\)   ...(82)
      
      \(K= D_1D_2\)   ...(83)
      
      Under such condition, the Value of \(C_1=E\), \(C_2=F\), \(D_1=G\) and \(D_2=H\) as given by equations (78), (79), (80) and (81) above respectively.
   6. If the Coefficients of Both \(x^2 \) (\(A\)), \(y^2 \) (\(B\)), \(z^2 \) (\(C\)) and \(xy \) (\(D\)) in the equation (1) are Zero and Coefficient of \(xz\) (\(E\)) in the equation (1) is Not Zero, then the Coefficient of \(xz\) (\(E\)) in the Normalized version of equation (1) becomes \(1\). Under such condition, one of the following 2 things can be done
      
      Either we can set \(A_1=C_2=1\) and \(A_2=C_1=0\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(84)
      
      \(B= 0\)   ...(85)
      
      \(C= 0\)   ...(86)
      
      \(D= B_2\)   ...(87)
      
      \(E= 1\)   ...(88)
      
      \(F= B_1\)   ...(89)
      
      \(G= D_2\)   ...(90)
      
      \(H= B_1D_2+B_2D_1\)   ...(91)
      
      \(I= D_1\)   ...(92)
      
      \(K= D_1D_2\)   ...(93)
      
      Under such condition, the Value of \(B_1=F\), \(B_2=D\), \(D_1=I\) and \(D_2=G\) as given by equations (89), (87), (92) and (90) above respectively.
      
      Otherwize, we can set \(A_1=C_2=0\) and \(A_2=C_1=1\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(94)
      
      \(B= 0\)   ...(95)
      
      \(C= 0\)   ...(96)
      
      \(D= B_1\)   ...(97)
      
      \(E= 1\)   ...(98)
      
      \(F= B_2\)   ...(99)
      
      \(G= D_1\)   ...(100)
      
      \(H= B_1D_2+B_2D_1\)   ...(101)
      
      \(I= D_2\)   ...(102)
      
      \(K= D_1D_2\)   ...(103)
      
      Under such condition, the Value of \(B_1=D\), \(B_2=F\), \(D_1=G\) and \(D_2=I\) as given by equations (97), (99), (100) and (102) above respectively.
      
      
   7. If the Coefficients of Both \(x^2 \) (\(A\)), \(y^2 \) (\(B\)), \(z^2 \) (\(C\)), \(xy \) (\(D\)) and \(xz \) (\(E\)) in the equation (1) are Zero and Coefficient of \(yz\) (\(F\)) in the equation (1) is Not Zero, then the Coefficient of \(yz\) (\(F\)) in the Normalized version of equation (1) becomes \(1\). Under such condition, one of the following 2 things can be done
      
      Either we can set \(B_1=C_2=1\) and \(B_2=C_1=0\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(104)
      
      \(B= 0\)   ...(105)
      
      \(C= 0\)   ...(106)
      
      \(D= A_2\)   ...(107)
      
      \(E= A_1\)   ...(108)
      
      \(F= 1\)   ...(109)
      
      \(G= A_1D_2 + A_2D_1\)   ...(110)
      
      \(H= D_2\)   ...(111)
      
      \(I= D_1\)   ...(112)
      
      \(K= D_1D_2\)   ...(113)
      
      Under such condition, the Value of \(A_1=E\), \(A_2=D\), \(D_1=I\) and \(D_2=H\) as given by equations (108), (107), (112) and (111) above respectively.
      
      Otherwize, we can set \(B_1=C_2=0\) and \(B_2=C_1=1\). Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      \(A= 0\)   ...(114)
      
      \(B= 0\)   ...(115)
      
      \(C= 0\)   ...(116)
      
      \(D= A_1\)   ...(117)
      
      \(E= A_2\)   ...(118)
      
      \(F= 1\)   ...(119)
      
      \(G= A_1D_2 + A_2D_1\)   ...(120)
      
      \(H= D_1\)   ...(121)
      
      \(I= D_2\)   ...(122)
      
      \(K= D_1D_2\)   ...(123)
      
      Under such condition, the Value of \(A_1=D\), \(A_2=E\), \(D_1=H\) and \(D_2=I\) as given by equations (117), (118), (121) and (122) above respectively.