Algebra Period 3 (Tuesday)

Quadratic Equations Review: Chapter 13


Finding the x intercepts of a parabola:
x intercepts = roots = solutions = zeros of a quadratic
You can find these one of two ways:
1) Read them from the graph

2) Set y or f(x) = 0 and then solve

Where the graph crosses the x axis is/are the x intercepts.

(Remember, y = 0 here!)
The x intercepts are the two solutions or roots of the quadratic.
When we factored in Chapter 6 and set each piece equal to zero,
We were finding the x value when y was zero.
That means we were finding these two roots!

You can find these roots (solutions, x intercepts, zeros) by several different methods.
You already know the following 2 methods:

1) Graphing and see where the graph crosses the x axis
2) Factoring, then using the zero products property to solve for x for each factor

ANOTHER METHOD:
Chapter 13-2 Solving Quadratics Using Square Roots -
You did this in Ch 11 for Pythagorean Theorem!
If there is no middle x term, it's easiest to just square root both sides to solve!

BUT THE DIFFERENCE FROM PYTHAGOREAN--NOT LOOKING FOR JUST THE PRINCIPAL SQUARE ROOT ANY MORE.

YOU NEED THE + OR - SYMBOL!!

EXAMPLE:
3x² = 18
divide both sides by 3 and get: x² = 6
square root each side and get x = + or - SQRT of 6

HARDER EXAMPLE:
(x - 5)² = 9
SQRT each side and get: x - 5 = + or - 3
+ 5 to both sides: x = 5 + 3 or x = 5 - 3
So, the 2 roots are x = 8 or x = 2

HARDEST EXAMPLE:
(x + 2)² = 7
SQRT each side and get: x + 2 = + or - SQRT of 7
-2 to both sides: x = -2+ SQRT 7 or x = -2 - SQRT 7

THE QUADRATIC FORMULA 13-4

-b plus or minus the square root of b squared minus 4ac all over 2a

Notice how the first part is the x value of the vertex -b/2a
The plus or minus square root of b squared minus 4ac represents
how far away the two x intercepts (or roots) are from the vertex!!!!

Very few real world quadratics can be solved by factoring or square rooting each side.
And completing the square always works, but it long and cumbersome!

All quadratics can be solved by using the QUADRATIC FORMULA.

(you will find out that some quadratics have NO REAL solutions, which means that there are no x intercepts - the parabola does not cross the x axis! Think about what kinds of parabolas would do this....ones that are smiles that have a vertex above the x or ones that are frowns that have a vertex below the x axis. You will find out in Algebra II that these parabolas have IMAGINARY roots)

So now you know 5 ways that you know to find the roots:

1. graph
2. factor if possible
3. square root each side
4. complete the square - that's what the quadratic formula is based on!
5. plug and chug in the Quadratic Formula -
This method always works if there's a REAL solution!

DON'T FORGET TO PUT THE QUADRATIC IN STANDARD FORM BEFORE PLUGGING THE VALUES INTO THE QUADRATIC FORMULA!

ax² + bx + c = 0

DISCRIMINANTS - a part of the Quadratic Formula that helps you to understand the graph of the parabola even before you graph it!
the discriminant is b² - 4ac

(the radicand in the Quadratic Formula, but without the SQRT)

Depending on the value of the radicand, you will know
HOW MANY REAL ROOTS IT HAS

1) Some quadratics have 2 real roots (x intercepts or solutions) - Graph crosses x axis twice
2) Some have 1 real root (x intercept or solution) - Vertex is sitting on the x axis
3) Some have NO real roots (no x intercepts or solutions) - vertex either is above the x axis and is a smiley face (a coefficient is positive) or
the vertex is below the x axis and is a frown face (a coefficient is negative)

In both of these cases, the parabola will NEVER CROSS (intercept) the x axis!

b² -4ac > 0 That is, if it's positive, then there are 2 roots
b² -4ac = 0 That is, if it's zero , then there is 1 root
b² -4ac <0>