Solving by graphing remediation notes

1. •Using Graphing

2. Investigating Solutions,
X-Intercepts and Factors

3. Solving Quadratics by Graphing

4. Types of Solutions
 There are 3 possible types of solutions evident
from the graph. Look at the number of times the
quadratic crosses the x-axis.
 2 REAL solutions when the quadratic has 2
different x-intercepts
 No REAL solution (2 complex roots) when the
quadratic has NO x-intercepts (can’t solve by
graphing-use completing the square or quadratic
formula!!)
We will discuss complex later!
 1 REALsolution when the quadratic has 1 x-
intercept, which is also the vertex
(a) 2 real (b) 2 complex (c) 1 real
y y
20 10

y
5 16 8
4 12 6
3
8 4
2
4 2
1 x x
x −20 −16 −12 −8 −4 4 8 12 16 20 −10 −8 −6 −4 −2 2 4 6 8 10
−5 −4 −3 −2 −1 1 2 3 4 5 -4 -2
-1
-8 -4
-2
-12 -6
-3
-16 -8
-4
-20 -10
-5

5. Examples:
Solve each quadratic using graphing.
 y = 4x² - 20 x  y = 3x² - 5x + 7
+25  Y1=3x²-5x+7
 Y1 = 4x²-20x+25  Y2=0
 Y2=0  NO intersection
 1 intersection (vertex)  No REAL solution
 x = 5/2 (1 real) 5
4
y
 2 complex solutions
y
3 20
2 16
12
1
x 8
−5 −4 −3 −2 −1 1 2 3 4 5
4
-1
x
-2 −20 −16 −12 −8 −4 4 8 12 16 20
-4
-3
-8
-4
-12
-5
-16
-20

6. You try: Find the real solutions
to the quadratic equation.
-x² - 4x + 6 = 0
 Click HERE to view the solution!

7.  Y1= -x²-4x+6 10
y
 Y2=0 8
6
 2 intersections 4
2
 x ≈ -5.16, 1.16 −10 −8 −6 −4 −2 2 4 6 8 10
x
-2
(2 real) -4
-6
-8
-10

8. View the following videos
for a review of solving by
graphing.
 http://www.phschool.com/atschool/academy123/html/bb
http://www.purplemath.com/modules/solvqu
ad5.htm
You may now take the Post Quiz to
Complete Module 7.