3.1 Notes: Solving Quadratic Equations By graphing and square roots

What does it mean to “solve?” When we are solving a quadratic equation, we are finding the x- intercept(s), real and non-real, that would be on our graph. There are at least five methods for solving, today we will look at two methods.

Solving by graphing Whether we graph by hand or on the graphing calculator, we are graphing a quadratic equation and looking for the place(s) where the graph touches the x – axis. Before we graph, we must make sure our equations are equal to 0!

Let’s try some: 1) x2 – x – 6 = 0 2) -2x2 – 2 = 4x

Solving by Square Roots We can solve by using backward order of operations (SADMEP), which would use square roots. The “b” term MUST be missing in order to do this (of ax2 + bx + c) ALL roots MUST be SIMPLIFIED! Don’t forget, negative roots mean we have to pull out “i”

Let’s try some: 1) 4x2 – 31 = 49 2) 3x2 + 9 = 0 3) (x + 3)2 = 5

Wait a minute, why couldn’t we … Why can’t we leave the radical in the denominator? It’s like having a decimal in the denominator…because it’s not a perfect square. We have to “rationalize” the denominator….which means multiplying the whole fraction by the non-perfect square of the denominator. For example: A) B) C)

HW: P. 99 #3 – 18

Warm UP: Factor completely (if possible): 1) 8n2 + 16n + 6 2) 2v2 + v – 5 3) -20x2 – 45

Solving by factoring: We all remember and LOVE factoring, right?  So, we can use factoring to help us solve quadratics! Remember, when we are solving quadratic equations, we are finding the x-intercepts!

Try this: First, let’s factor the following on our paper: X2 + 12x + 35

Now, let’s graph the quadratic in our graphing calculators: X2 + 12x + 35 = 0 What do you notice about the factored form we just found and the solution to the quadratic according to our graph?

Zero product property: So, the x-intercepts and the factored form we just found have the same values, but opposite signs. How does that happen? Well, what is the y value of the ordered pair of the x-intercept? Yes! This is why we have to make sure our quadratic equations are equal to zero before we start solving them, and why we can use the zero product property. The zero product property states that if the product of two expressions is zero, then one or both of the expressions is equal to zero.

Let’s try some together: Remember, our goal is to solve quadratic equations by factoring! On your white board, solve the following: 1) x2 – 8x + 12 = 0 2) 2x2 – 11x + 12 = 0 3) x2 – 4x = 45 4) x2 – 8x = 0 (what should this be equal to?) (What is the first thing we check for?) Remember, you may be able to check your answers on your graphing calculator, but we still need to see your work! 

Try these on your own: 1) 3x2 – 5x = 2 2) 4x2 + 28x + 49 = 0

Homework:P. 100 # 27 – 33, 47 – 54 (solve all by factoring Homework:P. 100 # 27 – 33, 47 – 54 (solve all by factoring!) SHOW ALL WORK!