 A quadratic expression or equation is a polynomial whose highest exponent is 2.
 The root root is the solution to the quadratic equation.
 Using factoring we find the solution of the quadratic equation.
 A quadratic will have a double root if the quadratic is a perfect square trinomial.
Factoring x^{2} + bx + c
x^{2} + bx + c = (x + p) (x + q) provided p + q = b and pq =c.
When factoring a trinomial, first consider the signs of p and q.
x^{2}+bx+c(x+2)(x+3) (x+2)(x+(3)) (x+(2))(x+3) (x+(2))(x+(3))
(x + p)(x + q)  signs of b and c 
x^{2}+5x+6  b is positive; c is positive 
x^{2}x6  b is negative; c is negative 
x^{2}+5x6  b is positive; c is negative 
x^{2}5x+6  b is negative; c is positive 
Observing the signs of b and c in the table we see that:
* b and c are positive when both p and q are positive.
* b is negative and c is positive when both p and q are negative.
* c is negative when p and q have different signs.
Examples:
 Factor x^{2} + 11x + 18
Find two positive factors of 18 whose sum is 11. Make a table
Factors of 18  Sum of factors  
18, 1  18+1=19  Not correct 
9,2  9 + 2 = 11  Correct sum 
6, 3  6 + 3 = 9  Not correct 
The factors 9 and 2 have a sum of 11, so they are correct values of p and q.
x^{2} + 11x + 18 = (x + 9)(x + 2)
 Show that the factors of n^{2}  6n + 8 is (n4)(n2)
Factors of 8  Sum of factors  
8, 1  8+(1)=9  Not correct 
4,2  4 + (2) = 6  Correct sum 
n^{2}  6n + 8 = (n4)(n2)
 Show that the factors of y^{2} + 2y  15 = (y + 5)(y  3)
Factors of 15  Sum of factors  
15, 1  15+1=14  Not correct 
15, 1  151=14  Not correct 
5, 3  5+3=2  Not correct 
5, 3  53=2  Correct sum 
y^{2} + 2y  15 = (y + 5)(y  3)
 Solve the equation x^{2} + 3x = 18
x^{2} + 3x = 18  original equation
x^{2} + 3x  18 = 0  subtract 18 from each side
(x + 6)(x  3) = 0  factor left side
(x + 6) = 0 or x  3 = 0
Therefore x = 6 or x = 3  solve for x
The solutions of the equation are 6 and 3
 Solve 6x^{2} + 42x = 0
6x^{2} + 42x = 0  original equation
6x(x + 7) = 0  factor left side
6x = 0 or x + 7 = 0  solve for x
x = 0 or x = 7
The solutions of the equation are 0 and 7
Directions: Solve the following problems. Also write at least ten examples of your own.
