9.4 - Quadratic equations

Before reading this section you may want to review the following topics: In this section we discuss four methods of solving quadratic equations:
  1. graphing,
  2. factoring,
  3. completing the square, and
  4. using the quadratic formula.

1. Solving quadratic equations by graphing

To solve the quadratic equation a x 2 + b x + c = 0, replace the zero on the right-hand-side with the variable y and graph the resulting quadratic function  y = a x 2 + b x + c. The graph is a parabola. The points where the parabola crosses the x axis are the points where y = 0 and hence are the roots or solutions of the quadratic equation.

This method is also the basis of computer methods used to solve more complicated equations.




Example: Solve the quadratic equation x 2 − 3 x + 2 = 0.

Solution: Replace 0 with y to create the corresponding quadratic function
y = x 2 − 3 x + 2
and draw its graph. The parabola crosses the x axis at x = 1 and x = 2. This means that the solutions of the quadratic equation x 2 − 3 x + 2 = 0 are x = 1 and x = 2.



2. Solving quadratic equations by factoring

The material in this section is based on the following topics, which you may want to first review: If the expression on the left-hand-side of the quadratic equation a x 2 + b x + c = 0 can be factored like this:

(xx1)(xx2) = 0,
then the solutions are x = x1 and x = x2. When you can spot the factors, this is probably the easiest of the four methods.



Example: Solve the quadratic equation x 2 + 3 = 4 x by factoring.

Solution: You must first put the quadratic equation into the standard form
x 2 − 4 x + 3 = 0.
The left-hand-side can be factored:
(x − 1)(x − 3) = 0.
Therefore the solutions of the quadratic equation are x = 1 and x = 3.


Warning: A common error is to think that the solutions are −1 and −3. This is not correct; the solutions are the values of x that make the factors vanish (become equal to zero).



3. Solving quadratic equations by completing the square

The quadratic equation a x 2 + b x + c = 0 can be solved for x by completing the square. Here are the steps:



Example: Solve the quadratic equation x 2 − 4 x + 3 = 0 by completing the square.

Solution: Here are the steps:




4. Solving quadratic equations by quadratic formula

The solutions of the quadratic equation a x 2 + b x + c = 0 are given by the quadratic formula:
Note:


Example: The solutions of the quadratic equation x 2 − 4 x + 3 = 0 can be found using the quadratic formula:

with a = 1, b = −4 and c = 3:
The + sign gives the solution x = 3 and the − sign gives the solution x = 1.




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