Solving Quadratic Equations
A quadratic equation is an equation of the form ,
ax2 + bx + c = 0 ,
where a, b, and c are real numbers , with a ≠ 0 . The condition, a ≠ 0 , ensures
that the
equation actually does have a x2- term .
Solving Quadratic Equations by Factoring
Zero Factor Theorem
If A and B are algebraic expressions , then the equation AB=0 is equivalent to
the
compound statement A = 0 or B = 0.
Examples:
Solve the quadratic equation, x2 + 2x −15 = 0 by factoring.
Solutions:
Using the zero factor theorem , we get
Thus, the solution set is {-5, 3}.
A Special Quadratic Equation
For any real number k , the equation x2 = k is equivalent to
.
If k > 0, then x2 = k has 2 real solutions.
k < 0, then x2 = k has no real solution.
k = 0, then 0 is the only solution to x2 = k .
Examples: Solve
(a) x2 = 361
(b) 16x2 = 49
Solutions:
Thus, the solution set for the equation x2 = 361 is {-19,
19} .
Thus, the solution set for the equation 16x2 = 49 is
Solving Quadratic Equations by Completing the Square
To complete the square of x 2 + kx , add
to both sides. That
is, add the
square of half the coefficient of x to both sides.
Examples:
Solve the fol lowing quadratic equations using completing the square.
(a) x2 − 8x +11 = 0
(b) 4x2 + 20x +13 = 0
Solutions:
(c) First, subtract 11 from both sides and then add
to both
sides:
The solution set is 
.
(d) Solve 4x2 + 20x +13 = 0 .
Solution: First, divide both sides by the leading coefficient, which is 4.
Now, subtract
from both
sides and add 
to both sides.
Thus, the solution set is
.
Solving Quadratic Equations using the Quadratic Formula
Quadratic Formula
The solution to ax2 + bx + c = 0 , with a ≠ 0 , is given by the formula,
provided b2 − 4ac ≥ 0 .
If b2 − 4ac > 0 , then there exists has 2 real roots .
b2 − 4ac < 0 , then there exist no real roots.
b2 − 4ac = 0 , then there exists one root (double root).
Example: Solve the quadratic equation, x2 +13x − 6 = 0 , using the quadratic
formula.
Solution: Since a = 5, b = 13, and c = −6 ,
Thus, the solution set is
.
