CHAPTER 4 - QUADRATIC EQUATIONS
NOTES
Quadratic Equation
any equation that can be written in the form , a 0
So a quadratic function and quadratic equation are very similar. Both have where
But a quadratic function is written
While a quadratic equation is written
A quadratic equation ()
- Can have two real roots (touches the x axis in two places)
- Can have one real root (touches the x axis in one place)
- Can have no real roots (does not touch the x axis)
Solving Quadratic Equations can be done in 4 ways
graphically (type equation into your calculator and find the x-intercepts, 2nd CALC ZERO)
factoring
vertex/completed square method (get to and solve for y = 0)
quadratic formula (will be taught in this unit. VERY POWERFUL!)
Graphically
A quick reminder of how to find x -intercepts on your TI calculator.
1.Get the equation to y = format or 0 = format.
2.Type the equation into your calculator by pressing y = (top left)
3.Change your WINDOW so that you can see any and all x -intercepts.
4.2nd / CALC / #2 Zero
Examples:
1.x2 + 3x = 10manipulate to
x intercept means y = 0 thus,
Method 1 (interesting, not needed)Method 2 (teach)
graph y = x2 + 3xgraph y = x2 + 3x + 10
graph y = 10use zero function (2nd CALC ZERO)
use intersect function
ANS: 5, 2
This means 2 real roots (2 real and different roots)
The graph has 2 different x-intercepts
Two different values for x makes the function = 0.
2.x2 8x + 16 = 0
ANS: 4
This means 1 real root
The graph has 1 x-intercept
Only 1 value for x makes the function = 0.
3.3x2 8x = 7
ANS: no real root
Manipulate to This means the graph does not cross the x axis.
No value for x makes this function = 0.
4.0.7x2 11x 17 = 0
ANS: x = 17.13, 1.42
This means 2 real roots (2 real and different roots)
The graph has 2 different x-intercepts
Two different values for x makes the function = 0.
5.2x2 + 47x = 0
ANS: 0, 23.5
This means 2 real roots (2 real and different roots)
The graph has 2 different x-intercepts
Two different values for x makes the function = 0.
6.Consider the equation
If you are asked for the x-intercepts, or how many roots or how may zeros it has, what would you do?
Would you move everything to the left?
Would you move everything to the right?
Does it matter?
Graph both equations and determine the x-intercepts, how many roots it has and how many zeros the equation has.
As you can see, both equations have 2 roots, 2 x-intercepts and 2 real zeros.
The direction of the graph is irrelevant if all you are concerning yourself with is what is happening on the x-axis.
Additional Questions
1.x2 + 5x = 11 ANS: 5.07, 1.53
2.3.8x2 4.6x + 2.1 = 1.7 ANS: no solutions
3.4(x + 1)2 = 1 ANS: 0.5,
M20-1 LP CH 4 Quad Eqn.docx