VISUAL ALGEBRA II represents a set of systematised algebraic factorisations and consists of 55 algebraic visualisations created to give meaning to:

• 5a1 Simple binomial factorisations of the following algebraic forms:
  • n a + n b   and   n a – n b,   [n = 2, 3, 4, 5];
• 5a2 Simple trinomial factorisations of the following algebraic forms:
  • n p + n q + n r   and   n p – n q + n r,   [n = 2, 3, 4];
• 5a3 Simple binomial factorisations of the form:
  • n p + 2 n q,   [n = 2, 3, 4];
• 5a4 Simple binomial factorisations of the form:
  • n a – 3 n b,   [n = 2, 3, 4];
• 5a5 Simple trinomial factorisations of the form:
  • n a + 2 n b + 3 n c,   [n = 2, 3, 4];
• 5a6 Simple trinomial factorisations of the form:
  • 2 n a – n b – 3 n c,   [n = 2, 3, 4];
• 5a7 Simple trinomial factorisations of the form:
  • 2 n a + n a x + 3 n a y,   [n = 2, 3, 4, 5, 6];
• 5a8 Simple trinomial factorisations of the form:
  • 3 n x + n x y – 2 n x z,   [n = 2, 3, 4, 5, 6];
• 5a9 Incomplete quadratic trinomials' factorisations of the form:
  • n x² + 3 n x,   [n = 1, 2, 3];
• 5a10 Incomplete quadratic trinomials' factorisations of the form:
  • 2 n x² – 3 n x,   [n = 1, 2, 3];
• 5b Two-stage factorisations of the following algebraic forms:
  • a c + a d + b c + b d,   a c – a d + b c – b d
  • - a c + a d – b c + b d,   a c + a d – b c – b d
  • a c – a d – b c + b d,   - a c + a d + b c – b d
  • - a c – a d + b c + b d,   - a c – a d – b c – b d;
• 5c1 Quadratic factorisations of the following algebraic forms:
  • x² + b x + c
  • x² – b x + c
  • x² – b x – c
  • x² + b x
  • x² – b x;
• 5c2 Difference of two squares' factorisations of the form:
  • x² – y²,   [y = 1, 2, 3, 4, 5, 6];
• 5c3 Full square factorisations of the following algebraic forms:
  • x² + 2 x y + y²   and   x² – 2 x y + y²,   [y = 1, 2, 3, 4, 5, 6];
• 5c4 Quadratic factorisations using surds of the following algebraic forms:
  • x² + 2 x – c   and   x² – 2 x – c,   [c = 1, 2, 3, 4, 5, 6, 7, 8, 9];

Applications of rational quadratic factorisations containing three equivalent forms of quadratic functions (quadratic circle) and explanations how to perform their conversions:

• 5d1   y = x² + 2 b x + b²   with a graph of parabola having its vertex on the border between 2^nd and 3^rd quadrants;
• 5d2   y = x² + b x + c   with a graph of parabola having its vertex in the 3^rd quadrant;
• 5d3   y = x² – c   with a graph of parabola having its vertex on the border between 3^rd and 4^th quadrants;
• 5d4   y = x² – b x – c   with a graph of parabola having its vertex in the 4^th quadrant;
• 5d5   y = x² – 2 b x + b²   with a graph of parabola having its vertex on the border between 4^th and 1^st quadrants;
• 5d6   y = - (x² – 2 b x + b²)   with a graph of upside-down parabola having its vertex on the border between 4^th and 1^st quadrants;
• 5d7   y = - (x² – b x – c)   with a graph of upside-down parabola having its vertex in the 1^st quadrant;
• 5d8   y = - (x² – c)   with a graph of upside-down parabola having its vertex on the border between 1^st and 2^nd quadrants;
• 5d9   y = - (x² + b x + c)   with a graph of upside-down parabola having its vertex in the 2^nd quadrant;
• 5d10   y = - (x² + 2 b x + b²)   with a graph of upside-down parabola having its vertex on the border between 2^nd and 3^rd quadrants;

• 5e Two-step, two-stage factorisations expressed in the form:
  • n (± a ± b) (± c ± d),   [n = - 3, - 2, - 1, + 1, + 2, + 3];
• 5f Two-step, two-stage quadratic factorisations expressed in the form:
  • n (± x ± a) (± x ± b),   [n = - 3, - 2, - 1, + 1, + 2, + 3];

Two-step, two-stage quadratic factorisations having a common factor and expressed in the form:

• 5g1a   n (x² + b x + c),   [n = 1, 2, 3, 4, 5, 6];
• 5g1b   n (x² – b x – c),   [n = 1, 2, 3, 4, 5, 6];
• 5g1c   n (x² + b x – c),   [n = 1, 2, 3, 4, 5, 6];
• 5g1d   n (x² – b x + c),   [n = 1, 2, 3, 4, 5, 6];
• 5g2   n (x² – y²),   [n = 1, 2, 3, 4, 5, 6];
• 5g3   n (x² ± 2 x y + y²),   [n = 1, 2, 3, 4, 5, 6];

Two-step, two-stage quadratic factorisations not having a common factor and expressed in the form:

• 5h1   a x² + b x + c   with both horizontal and vertical grouping of terms;
• 5h2   a x² + b x – c   with both horizontal and vertical grouping of terms;
• 5h3   a x² – b x – c   with both horizontal and vertical grouping of terms;
• 5h4   a x² – b x + c   with both horizontal and vertical grouping of terms;
• 5h5    a x² + b x + c   visualised as four basic linked examples;
• 5h6    a x² – b x – c   visualised as four basic linked examples;
• 5h7    a x² + b x – c   visualised as five basic linked examples;
• 5h8    a x² – b x + c   visualised as five basic linked examples;
• 5h9    a x² + b x + c   visualised as four sophisticated linked examples;
• 5h10    a x² + b x – c   visualised as four sophisticated linked examples;
• 5h11    a x² – b x – c   visualised as four sophisticated linked examples;
• 5h12    a x² – b x + c   visualised as four sophisticated linked examples;
• 5h13    a x² + b x + c   visualised as three linked more sophisticated examples;
• 5h14    a x² – b x + c   visualised as three linked more sophisticated examples;
• 5h15    a x² – c   visualised as three linked more sophisticated examples;
• 5h16    a x² + b x   visualised as examples of incomplete quadratic trinomials;
• 5h17    a x² – b x   visualised as examples of incomplete quadratic trinomials;

Three-stage quadratic factorisations expressed in the form:

• 6a    n (a x² + b x + c)   visualised as factorised examples   n (d x + e) (f x + g);
• 6b    n (a x² ± b x – c)   visualised as factorised examples   n (d x – e) (f x + g);
• 6c    n (a x² ± b x – c)   visualised as factorised examples   n (d x + e) (f x – g);
• 6d    n (a x² – b x + c)   visualised as factorised examples   n (d x – e) (f x – g).