In this section, we discuss various ways for solving the
equation ax2+bx+c =
0. A solution, also known as a root or a zero, is a number satisfying the
equation.
Factored Form of a Quadratic Function
The form f(x) = ax2+bx+c is knwon as the standard form of a quadratic
function. Another form is the so-called factored form given by
The factored form is obtained from the standard form by
applying the fol-
lowing factoring process.
1. find two integers that have a product equal to ac and a sum equal to b;
2. replace bx by two terms using the two new integers as coefficients ,
3. then factor the resulting four-term polynomial by grouping . Thus, ob-
taining (a1x - b1)(a2x - b2)
Once a quadratic function is factored as f(x) = (a1x
-
b1)(a2x - b2), then
the zeros of the equation f(x) = (a1x - b1)(a2x
- b2) = 0 are the horizontal
intercepts of f . They are the solutions to the two equation a1x - b1 = 0 or
a2x - b2 = 0. We illustrate this in the next example.
Example 1
Find the zeros of f(x) = x2 - 2x - 8 using factoring.
Solution.
We need two numbers whose product is -8 and sum is -2. Such two integers
are -4 and 2. Thus,
Thus, either x = -2 or x = 4.
Solving by Completing the Square
We can solve a quadratic equation by completing the square as discussed in
Section 21.
Thus, 
If a and k are
of the same sign then
the right -hand side is negative whereas the left-hand side is positive or zero ,
which is impossible so we say that the equation has no solutions. If a and k
are of opposite sign then there are two solutions given by
Finding the x in the last step was based on the so-called
the square root
method.
Example 2
Solve.
(a) (x + 4)^2 = -1
(b) (x + 4)^2 = 1.
Solution.
(a) The left-hand side is positive or zero whereas the right-hand side is neg-
ative. The equation has no solutions.
(b) We have (x + 4)^2 = 1. Taking square root of both sides to obtain
x + 4 = ± 1 and solving for x we find the two solutions x = -4 ± 1. That is,
x = -4 + 1 = -3 and x = -4 - 1 = -5
Example 3
By completing the square solve. f(x) = -4x2 - 12x - 8 = 0.
Solution.
Using the method of completing the square we find
Thus,
Since the
left-hand side is positive
and the right-hand side is negative, the equation has no solutions
Solving Using the Quadratic Formula
The quadratic formula is the result of the method of completing the square
as can be seen below.
The formula as you can see involves a square root. Thus,
for this formula
to make sense we must have b^2 - 4a≥0. If b^2 - 4ac < 0 then the quadratic
equation is said to have no roots (in fact it has roots known as complex roots,
a topic not covered in this course). If b^2 -4ac > 0 then the equation has the
two distinct roots
Finally, if b^2 - 4ac = 0 then the equation has a
repeated root 
We
call the ex pression D = b^2 - 4ac the discriminant.
Example 4
How many solutions does each of the following equation have? You do not
need to find the solutions.
(a) 2x2 - x - 1 = 0
(b) x2 - 4x + 4 = 0
(c) x2 + x + 1 = 0.
Solution.
The number of solutions depends on the sign of D.
(a) Here we have a = 2; b = -1; c = -1 so that D = (-1)^2-4(2)(-1) = 9 > 0
which implies two different solutions .
(b) We have a = 1; b = -4; c = 4 so that D = (-4)^2 - 4(1)(4) = 0. One
single solution.
(c) We have a = b = c = 1 so that D = 1^2 - 4(1)(1) = -3 < 0. No real
solutions
Example 5
Solve each of the following equations.
(a) 2x2 - x - 1 = 0
(b) x2 - 4x + 4 = 0
(c) x2 + x + 1 = 0.
Solution.
(a) From (a) above we found D = 9 so that the two different roots are
(b) From (b) we have D = 0 so that the only root is
(c) Since D = -3 < 0; the equation has no real solutions
Graphical Method
Quadratic equations can be solved graphically. One enters the equations
y = ax2 +bx+c and y = 0 into the calculator and use the key INTERSECT
to nd the x-intercepts. We will not pursue further graphical methods in
this section.
