Dec 1 : 'Use the distributive property of multiplication over addition or subtraction to change the form of expression to take off parentheses'. Dec 2 : 'Rearrange the equation in descending power of one variable'. Dec 3 : 'Multiply the least common multiple of the denominators in order to simplify the equation with integers'. Dec 4 : 'It cannot be considered as a quadratic form'. Dec 5 : 'It is a linear equation'. Dec 6 : 'Use the values T and T^2 as substitutes of x^2 and x^4, consecutively and consider it as a new quadratic equation.'. Dec 7 : 'The solution is x=0'. Dec 8 : 'Use inverse operations to write an equivalent equation in the form "variable = number" i.e. the variable terms on the left side and the constant on the right side, and divide each side by the coefficient of the second order term'. . . . . Dec 9 : 'Extract the root in order to obtain the solution x=+sqrt(c/a) and x=-sqrt(c/a)'. Dec 10 : 'There is no solution in real numbers'. Dec 11 : 'Check if the solution satisfy the equation'. Dec 12 : 'Take out any common factor ,especially the variable x out of all x-terms, except for 1'. Dec 13 : 'Apply the Zero product property into the equation "If ab=0 then a=0 or b=0 i.e. the product of two polynomials is zero providing one or both of the polynomials is equal to zero"'. Dec 14 : 'Solve the resulting linear equation'. Dec 15 : 'Find the factors (m and n) of the first term and the factors (p and q)of the last term, and try all possible combination until you find the correct middle term, which means the combination "mq+np" is equal to the middle term'. Dec 16 : 'Use the quadratic formula x=(-b +-sqrt(b^2-4ac))/2a'. Dec 17 : 'Factor the equation using (x+m)(x+n)= x^2+(m+n)x+mn '. Dec 18 : 'Factor the trinomial to make a complete square using (x+y)^2=x^2+2xy+y^2 or (x-y)^2=x^2-2xy+y^2'. Dec 19 : 'Find two numbers m and n whose product is the last term and whose sum is the middle term'. Dec 20 : 'Let B=b/2 and use the formula x=(-B+-sqrt(B^2-ac))/a'. Dec 21 : 'Factor the equation using this patttern (mx+p)(nx+q)=mnx^2+(mq+np)x+pq'. Dec 22 : 'Factor a binomial that is the difference of two squares by using the difference of squares multiplication pattern A^2-B^2=(A+B)(A-B) '. Dec 23 : 'Rewrite the equation to make standard form ax^2+bx+c=0'. Dec 24 : 'Take out the negative common factor'. Dec 25 : 'Replace T with the solution (because you changed variable x^2=T and x^4=T^2) and solve the equations with the variable x'. Question 1 : 'Does the equation have parentheses?' Why '' Answers 1 'Yes' 2 'No' . Question 2 : 'Is the equation in descending power of one variable?' Why 'Even though there are several variables, we are going consider only one variable' Answers 1 'Yes' 2 'No'. Question 3 : 'Does the equation have a rational expression except for denominator 1?' Why 'It is better to simplify the expression with integers in order to easily solve it ' Answers 1 'Yes' 2 'No'. Question 4 : 'What is the highest degree of the equation?' Why '' Answers 1 '5 more' 2 '4' 3 '3' 4 '2' 5 '1'. Question 5 : 'Is the form of equation ax^4+bx^2+c=0 where a, b, and c are real numbers and a is not 0 ?' Why '' Answers 1 'yes' 2 'no'. Question 6 : 'Is the coefficient of the first order term 0?' Why '' Answers 1 'Yes' 2 'No'. Question 7 : 'Is the constant term 0?' Why '' Answers 1 'Yes' 2 'No'. Question 8 : 'Can you recognize the pattern of the difference of two squares "(A^2-B^2)" in your equation?' Why '' Answers 1 'Yes' 2 'No'. Question 9 : 'In x^2=c/a, what is the sign of the c/a' Why '' Answers 1 'c/a>0' 2 'c/a<0'. Question 10 : 'Is the equation the standard form of a trinomial ax^2 + bx + c = 0 where a, b, and c are real numbers ?' Why '' Answers 1 'Yes' 2 'No'. Question 11 : 'Is the leading coefficient of a trinomial negative?' Why '' Answers 1 'Yes' 2 'No'. Question 12 : 'Can you find two integers m and n, whose product is equal to (the coefficient of the second order term)*(the constant term) and whose sum is equal to the coefficient of the first order term' Why 'If you can, the equation is factorable with integer coefficients' Answers 1 'Yes' 2 'No'. Question 13 : 'What is the sign of b^2-4ac?' Why '' Answers 1 'b^2-4ac>=0' 2 'b^2-4ac<0 '. Question 14 : 'Is b even?' Why '' Answers 1 'Yes' 2 'No'. Question 15 : 'Can you recognize the perfect square pattern x^2 + 2xy + y^2 = (x+y)^2 or x^2 - 2xy + y^2 = (x-y)^2 in your equation?' Why '' Answers 1 'Yes' 2 'No'. Question 16 : 'Is the leading coefficient of the trinomial 1 ?' Why '' Answers 1 'Yes' 2 'No'. Rule 1 : Why '' IF q1a1 THEN d1 . Rule 2 : Why '' IF (q1a2 or d1) and q2a2 THEN d2 . Rule 3 : Why '' IF (q1a2 or d1) and (q2a1 or d2) and q3a1 THEN d3 . Rule 4 : Why '' IF (q1a2 or d1) and (q2a1 or d2) and (d3 or q3a2) and (q4a1 or q4a3) THEN d4 . Rule 5 : Why '' IF (q1a2 or d1) and (q2a1 or d2) and (d3 or q3a2) and q4a2 and q5a1 THEN d6 . Rule 6 : Why '' IF (q1a2 or d1) and (q2a1 or d2) and (d3 or q3a2) and q4a5 THEN d5 . Rule 7 : Why '' IF d5 THEN d14. Rule 8 : Why '' IF (q1a2 or d1) and (q2a1 or d2) and (d3 or q3a2) and q4a2 and q5a2 THEN d4 . Rule 9 : Why '' IF (d6 or q4a4) and q6a1 and q7a1 THEN d7 . Rule 10 : Why '' If d6 and d9 Then d25 Rule 11 : Why '' IF (d6 or q4a4) and q6a1 and q7a2 and q8a2 THEN d8. Rule 12 : Why '' IF (d6 or q4a4) and q6a1 and q7a2 and q8a1 THEN d22. Rule 13 : Why '' IF d22 THEN d13 . Rule 14 : Why '' IF d13 THEN d14 . Rule 15 : Why '' IF d14 THEN d11. Rule 16 : Why '' IF d6 and d11 THEN d25. Rule 17 : Why '' IF d8 and q9a1 THEN d9 . Rule 18 : Why '' IF d8 and q9a2 THEN d10 . Rule 19 : Why '' IF (d6 or q4a4) and q6a2 and q7a1 THEN d12 . Rule 20 : Why '' IF d12 THEN d13 . Rule 21 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and q10a2 THEN d23 . Rule 22 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and q11a1 THEN d24 . Rule 23 : Why '' IF d23 and q11a1 THEN d24 . Rule 24 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a2 and q13a2 THEN d10 . Rule 25 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a2 and q13a1 and q14a1 THEN d20 . Rule 26 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a2 and q13a1 and q14a2 THEN d16 . Rule 27 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a1 and q15a1 THEN d18 . Rule 28 : Why '' IF d18 THEN d13 . Rule 29 : Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a1 and q15a2 and q16a1 THEN d19 . Rule 30 : Why '' IF d19 THEN d17 . Rule 31 : Why '' IF d17 THEN d13 . Rule 32: Why '' IF (d6 or q4a4) and q6a2 and q7a2 and (q10a1 or d23) and (q11a2 or d24) and q12a1 and q15a2 and q16a2 THEN d15 . Rule 33 : Why '' IF d15 THEN d21 . Rule 34 : Why '' IF d21 THEN d13 .