Download Solving Quadratics Rules

FACTORING
(GCF, Difference of Squares, Difference of Cubes, Sum of Cubes,
Trinomial shortcut, Split the Middle, Grouping)
To Solve:
1. Get the equation = 0
2. Factor by one of the above methods
SEE INDIVIDUAL SHEETS FOR DIFFERENT FACTORING RULES
3. Set every factor = 0 in their own equation
4. Solve each equation
GCF FACTORING
1. List the prime factors of each term independently
2. Factor out any term that is in common with ALL terms
3. Write the remaining terms as a polynomial
4. Try to factor the remaining polynomial
EXAMPLE
4 x3  12 x 2  6 x
4 x3  2*2* x * x * x
12 x 2  2*2*3* x * x
6 x  2*3* x
(2 x)(2 x 2  6 x  3)
Difference of Squares
FACTORING
1. Try to find a GCF, factor out if there is one
2. Determine if both terms are perfect square
3. If they are, draw two sets of parentheses
4. Put a plus (+) in the first and a minus (-) in the second
5. The first term in both parentheses will be the square root of term 1
6. The second term in both will be the square root of term 2
7. Check to see if either set factors more
EXAMPLE
9 x 2  25
** no GCF **
9 x 2  3x
25  5
(3 x  5)(3 x  5)
Sum/Difference of Cubes
FACTORING
1. Try to find a GCF, factor out if there is one
2. Determine if both terms are perfect cubes
3. If they are, draw two sets of parentheses
4. Set 1 is a binomial and set 2 is a trinomial
5. Use S.O.A.P. to find your signs
s.o.a.p. stand for Same, Opposite, and Always Positive
5. The terms in the binomial are the cube roots of term 1 and term 2
6. The 1st term in the trinomial is the square of your binomial’s 1st term
7. The 2nd term in the trinomial is the product of the binomial terms
8. The 3rd term in the trinomial is the square of your binomial’s 2nd term
EXAMPLE
x3  27
** no GCF **
3
x3  x
27  3
S .O. A.P (_  _)(_  _  _)
3
( x  3)( x 2  3 x  9)
( x) 2  x 2
( x)(3)  3 x
(3) 2  9
Trinomial Shortcut
FACTORING
THIS ONLY WORKS IF THERE IS A 1 IN FRONT OF X2
1. Find all the integers that multiply to give you the constant
2a. If the constant is positive, add the numbers
2b. If the constant is negative, subtract the numbers
3. Choose the pair that add/subtract to equal your middle term
4. The first term in each parentheses will be x
5. The second term will be the two numbers you chose in step 3
BE SURE TO KEEP THE NEGATIVE SIGN IF SUBTRACTING
EXAMPLE
x 2  3 x  28
 28  (1* 28)  (2*14)  (4*7)
subtract (28  1  27) (14  2  12) (7  4  3)
choose 7 and  4
( x  7)( x  4)
Split the Middle
FACTORING
TRY TO FIND A GCF FIRST
1. Find the “key” by doing a*c
2. Find the multiples of the key
3. Add/Subtract the multiples of the key
4. Choose the pair that results in the middle number
5. Re-write the equation as 4 terms using your two new middle terms
6. Group together term1 and term2, then term3 and term4
7. Pull out GCF’s for both pairs
8. Resulting parentheses should be the same
9. First parenthesis is going to be the common one from step 8
10. Second parenthesis will be what is in front of the common pair
EXAMPLE
10 x 2  23x  12
key  10*12  120
(8) (15)  23
10 x 2  8 x  15 x  12
10 x 2  2*5* x * x
15 x  3*5* x
8 x  2* 2* 2* x
12  2* 2*3
(2 x)(5 x  4)
(3)(5 x  4)
(5 x  4)(2 x  3)
Grouping
FACTORING
1. Group the terms into two groups of two
2. Be sure that each pair will have a GCF
3. Pull out GCF’s for both pairs
4. Resulting parentheses should be the same
5. First parenthesis is going to be the common one from step 4
6. Second parenthesis will be what is in front of the common pair
EXAMPLE
14 x 2  2 x  35 x  5
14 x 2  2*7 * x * x
35 x  5*7 * x
 2 x  1* 2* x
 5  1*5
(2 x)(7 x  1)
(7 x  1)(2 x  5)
(5)(7 x  1)
QUADRATIC FORMULA
b  b2  4ac
2a
To Solve:
1. Set the equation = 0
2. Find a, it is the number in front of x2
3. Find b, it is the number in front of x
4. Find c, it is the number that does not have a variable
BE CAREFUL OF THE HIDDEN COEFFICIENT OF 1
5. Plug all of the values into the equation
6. Calculate
7. Reduce all radicals
COMPLETING THE SQUARE
To Solve:
1. Clear the constant to the right and move all variables to the left
2. If there is a number in front of x2, divide all terms by that number
3. Place an empty box on each side of the equation
4. Divide the number in front of x by 2 and square the answer
5. Put this answer in the box
6. Factor the P.S.T. on the left and do the addition on the right
7. Square Root both sides
BE SURE TO INCLUDE A POSITIVE AND NEGATIVE ON THE RIGHT
8. Create two equations by using the positive and then the negative
9. Solve each equation
BE SURE TO SIMPLIFY ALL RADICALS