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Solving Polynomial Equations by Graphing

Types of Equations Quadratic - has the form ax2 + bx + c = 0 Highest exponent is two (this is the degree) The most real solutions it has is two.

Types of Equations Cubic - has the form ax3 + bx2 + cx + d = 0 Highest exponent is three (this is the degree) The most real solutions it has is three.

Types of Equations Quartic - has the form ax4 + bx3 + cx2 + dx + e = 0 Highest exponent is four (this is the degree) The most real solutions it has is four.

Types of Equations These keep on going up as the highest exponent increases. You don’t need to know the names above quartic, but you do need to be able to give the degree.

Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. Instead of ‘solve’ we call this finding ‘zeros’ or ‘roots’.

Solving Equations Get all the x or constant terms on one side.
If you have a y or f(x), replace it with 0.

Solving Equations The first way we are going to solve these equations is by graphing. (Yeah!!! More calculator stuff!!) Go to the graph menu on your calculator.

Solving Equations Solve: x2 - 4 = y Replace y with 0.
Plug in x2 - 4 into your calculator. Graph it and let’s look at the graph.

Solving Equations When we talk about the graph and we are looking for places where y = 0, where will these points be? On the x-axis. So we are looking for the x-intercepts.

Solving Equations So where does this graph cross the x-axis?
(2, 0) and (-2, 0) If you can’t tell from looking at the graph, go to F5 (gsolv) and then F1 (root).

Solving Equations This should give you the first zero, to get to the second, hit the right arrow button. Note: the zeros should be on the screen. If you can’t see the x-intercepts, make your window bigger.

Solving Equations So the solutions to this equation are x = 2 or x = -2.

Solving Equations Find the solutions to f(x) = x2 - 5x + 6. x = 3, 2
Find the zero’s of 0 = x2 - 4x + 4 x = 2

Solving Equations How do we check our solutions?
Plug in and see if the equation simplifies to 0.

Solving Equations Let’s look at quadratic equations for a minute.
How many solutions should you look for? Two, one or zero.

Solving Equations Let’s look at some cubic equations. x3 - 1 = 0 x = 1
x3 - 6x + 1 = f(x) Has three solutions.

Solving Equations When we are solving cubic equations, we will have either 3, 2, or 1 real solution. You should never have no solutions.

Solving Equations What about quartic equations? They look like W or M.
They could have four, three, two, one, or no solution.

Solving Equations Let’s look at your graphing equations worksheet.

Factoring For these last two methods for solving equations, we will be looking at only quadratic equations (degree 2). The next method we will look at is factoring.

Factoring Quadratics We know that quadratic equations are set equal to 0. We will factor the trinomial and set each factor equal to 0 to find our solutions.

Factoring Quadratics x2 - 4 = 0
Let’s try the first graphing example and factor it. to factor x2 - 4 we use difference of squares. x2 - 4 = (x - 2)(x + 2) = 0

Factoring Quadratics Okay, let’s take a side note for a second.
If we multiply two numbers and get a product of 0, what do the factors have to be? 3x = 0, what does x have to be?

Factoring Quadratics if ab = 0, what do we know about a or b.
Either a has to be 0, b has to be 0, or they both can be zero. This is the only way to get a product of 0.

Factoring Quadratics Okay, back to factoring. (x - 2)(x + 2) = 0
So x - 2 = 0, meaning x = 2 or x + 2 = 0, meaning x = -2 So our solutions are x = 2 and x = -2.

Factoring Quadratics Find the roots by factoring: 2x2 + 8x - 24 = 0
First, factor 2x2 + 8x - 24. 2(x + 6)(x - 2). Set each factor (that contains an x) equal to zero.

Factoring Quadratics x + 6 = 0 x = -6 x - 2 = 0 x = 2
So x = -6 or x = 2.

Quadratic Formula The last method we will use to solve quadratic equations is the quadratic formula. This is the only method that will ALWAYS work when trying to solve a quadratic equation.

Quadratic Formula All the quadratic formula is is plugging in numbers.
You don’t need to worry about memorizing it. They give it to you on the SOL

Quadratic Formula Let’s look back the the general form of a quadratic equation.

Quadratic Formula ax2 + bx + c = 0
a is the coefficient of the squared term. b is the coefficient of the x term. c is the constant.

Quadratic Formula If one of these three terms doesn’t exist, then the coefficient of that term will be ____?

Quadratic Formula what is the quadratic formula?

Quadratic Formula Let’s look at an example. 3x2 - 4x + 3 = 0 a = ?
b = ? c = ? a = 3 b = -4 c = 3

Quadratic Formula Now let’s plug it in. b = -4, so -b = -(-4) = 4

Quadratic Formula Simplify

Quadratic Formula Keep going now.

Quadratic Formula

Quadratic Formula Find the zeros of r2 - 7r -18 = 0

Quadratic Formula Find the zeros of r2 - 7r -18 = 0 a = 1 b = -7 c = -18

Quadratic Formula Find the zeros of r2 - 7r -18 = 0

Quadratic Formula Simplify

Quadratic Formula Now let’s examine our solution. We can break this into two equations.

Quadratic Formula Now we can get our two solutions.

Quadratic Formula Now you try some. pg. 357 (only solve them using the quadratic formula)

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