Algebra (Period 4)

More on Solving Quadratics 13-2


CHAPTER 13 gives you several different ways to SOLVE QUADRATICS


Solving a quadratic means to find the x intercepts of a parabola.

There are different ways of asking the exact same question:

Find the:
x intercepts = the roots = the solutions =
the zeros of a quadratic


We'll answer this question one of the following ways:


1) Read them from the graph (read the x intercepts)
that's CHAPTER 12-4 that we just did!

But...
Graphing takes time and sometimes the intercepts are not integers so they'd be hard to read exactly


2) Set y or f(x) = 0 and then factor (we did this in Chapter 6 & 13-1)


but...some quadratics are not factorable


3) Square root each side (from Chapter 11 & 13-2)

Remember that in this case, you will need both the + or - square roots unlike the Pythagorean Theorem when we just wanted the positive square root since it was a side of a triangle.


But... sometimes the variable side is not a perfect square



4) If not a perfect square on the variable side, we'll learn to do a process called "completing the square", (Chapter 13-3) (Next Week!!)

then solve using #3 method


Now this method ALWAYS works,

But...it takes a lot of time and can get complicated




5) Quadratic Formula (Chapter 13-4) Works for EVERY quadratic!


Really easy if you just memorize the formula and learn how to use it! :)




REVIEW OF CHAPTER 13-1: Reading the x intercepts from a graph or factoring and solving using the zero products property.




METHOD 1:

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. 




METHOD 2:

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! 




METHOD 3:

Chapter 13-2: Use Square Roots -

You did this in Chapter 11 for Pythagorean Theorem!


If there is no x term, it's easiest to just square root both sides to solve!




DIFFERENT FROM PYTHAGOREAN: NOT LOOKING FOR JUST THE PRINCIPAL SQUARE ROOT ANYMORE. NEED THE + OR - SYMBOL!!!



This is also different from what we did when we had radical equations. When we did that, we SQUARED both sides to solve it.

It ended up looking like these---> but only after we squared both sides!
The reason we had to check both sides for the solution in that case is that we had changed the equations by squaring them---> that is NOT the case here, BUT you should always check your solutions for mistakes.



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