Objective:
Solving a quadratic equation symbolically (by factoring), numerically (using
table) and
graphically (using a graph).
Solving Symbolically
Remember to solve a quadratic by factoring , follow these steps .
Step 1: If necessary, write the equation in ____________________ form
ax2 +bx + c = 0
Step 2: ________________ the quadratic ex pression of the equation using
any method.
Step 3: Apply the __________-___________ rule .
Step 4: Solve each of the resulting __________________. Check any
________________.
1. Solve the equation symbolically: x2 −16 = 0
2. Solve the equation symbolically: x2 −6x = −9
3. Solve the equation symbolically: 3b2 = 2−b
4. Solve the equation symbolically: 4x2 +1= 0
Solutions to each of the previous equations could have
been obtained by numerical and
graphical means , which will be explored on the next pages.
Recall: To solve an equation using a table, you
look for the row where the outputs are equal.
The solution is the x- value of that row .
Solving Numerically using a Graphing Calculator
Step 1: If desired, write the equation in ____________________ form ax2 +bx
+ c = 0
Note: It is not necessary to write the equation in this form, but often is
useful.
Step 2: In
type the quadratic expression.
Step 3: View the table using
and look for the row(s) where the output is zero. If
there is no output of zero on the table , then look in between rows of the table
where the output
changes from negative to positive . Do this by changing the table setup:
to get
desired accuracy.
Step 4: The solution(s) to the equation is the x-value of the row. Copy
the row above and below
your solution on your paper as your work.
For problems 5 – 8, solve each equation numerically.
Copy the row above and below your
solution on your paper as your work.
5. x2 −16 = 0
Solutions: x = ____ and x = ____
6. x2 −6x = −9
7. 3b2 = 2−b
8. 4x2 +1= 0
Note: When graphing a two variable quadratic
equation y = ax2 + bx + c , the graph will be a U
shaped curve, called a parabola . This graph will be studied more in Chapter 9.
For now, we
will use our graphing calculators to graph these equations. It is useful to move
the expression to
the side that makes the quadratic term, ax2 , positive.
Solving Graphically using a Graphing Calculator
Step 1: Write the equation in ____________________ form ax2 +bx + c = 0
Note: Again it is not necessary to write the equation in this form, but often is
useful. If not
written in this form, a Y1 and Y2 must be used, as well as
select 5: INTERSECTION
Step 2: In
type the quadratic expression and set a good
so that you can see the
parabola, and, in particular, the points where the graph crosses the x-axis.
Step 3: Use
;
select 2: ZEROS, to find the x- intercepts of the graph . Follow the
prompts to give LEFT and RIGHT BOUNDS and a GUESS for each x-intercept.
Step 4: The solution(s) to the equation is the x-coordinate of any zero
(or x-intercept) of the
graph. Sketch a graph of the curve and label the solutions on the graph.
9. Solve the equation graphically:
10. Solve the equation graphically:
11. Solve the equation graphically:
12. Solve the equation graphically:
Putting it All Together:
13. Solve the quadratic equation 4x2 − 25x = 21 symbolically, numerically
and graphically.
Polynomial
Expressions versus Equations
While there are similarities between your treatment of
polynomial expressions and polynomial
equations, you must keep focused on their differences as well .
Polynomial Expressions:
DO
NOT set the expression equal to zero.
DO
write the expression in standard form ( descending powers ).
DO
look for the GCF when factoring.
DO
factor completely, using appropriate factoring techniques.
DO
NOT set each factor equal to zero.
DO
NOT solve for x.
DO
leave your polynomial as a factored expression
14. 5x3 − 20x2 − 25x
15. −5a3 + 45a
16. 8x3 − 24x2 +18x
17. 81a4 +18a2
Polynomial Equations:
DO
set the equation equal to ZERO.
DO
write the polynomial in standard form.
DO
look for the GCF when factoring.
DO
factor completely, using appropriate factoring techniques.
DO
set each factor equal to zero.
DO
solve for x from each factor.
DO
NOT leave your solution as a factored expression.
18.5x2 −12 = −17x
19. x(8x − 2) = 6
