Today:

Warm-Up: review for test tomorrow,

Review: ax2 + bx + c trinomials

Solving Quadratic Equations

The graph of a parabola

Class Work

Part I of our 3rd quarter final will be on new material,

including ax2 + bx + c trinomials & Quadratic Equations

Part II will cover all other 3rd quarter concepts.

1) Parallel & Perpendicular Lines

2) Systems of Equations & Inequalities

3) Monomials: Adding, Subtracting, Multiplying, Dividing

4) Scientific Notation

5) Polynomials: Adding, Subtracting, Multiplying, Dividing

6) Factoring Polynomials

Warm-Up:

Factoring Test I Tomorrow:

(x3 - 2x2 + 3x - 6)

(4x3 + 24x2 + 36x)

1) GCF 2) Grouping 3) Factor x2 + bx + c Trinomials

4) Special Products 5)Perfect Square Trinomials

Let's Meet the Enemy: 4x(x + 3)2

(x – 2)(x2 + 2) (x2 - 15) Prime

(4x2 + 64) 4(x2 + 18) x2 - 11x + 28 (x - 7)(x – 4)

6x - 18 12x - 163x2 + 2x - 8 4x - 12

•

Warm-Up:

35x2 + 2x - 24 Our signs will be... ( ) ( )+ -

Possible factors of 35x2 are {x, 35x} or {5x, 7x}

The factors of 24 could have either sign, so both must be tested. (1, -24) or (-1, 24) (2, -12) or (-2, 12), etc.

Since the middle term is a small number, the factors must be close to each other. Remember this important point.

Factor 2x – 9y + 18 – xyFactor : -4x 2 + 19x – 21

1. If the highest degree of a quadratic equation is 2, why are they called 'quadratic'?

Good Question, thanks for asking. Quadratic Equations were used as early as 1500 b.c., by Egyptian farmers.

x2 = 25, 4y2 + 2y - 1 = 0, y2 + 6y = 0, x2 + 2x - 4 = 0

The standard form of a quadratic is written as: ax2 + bx + c = 0, where only a cannot = 0

The name came much later 500 a.d. during Roman empire. If something is raised to the 2nd degree is referred to as 'squared', so Quad it became.

Quadratic Equations

The following are all examples of quadratic equations:

Quadratic EquationsA). The graphs of quadratics are not straight lines, they are always in the shape of a Parabola.

B) Parabolas ALWAYS have two solutions.

C) The slope of a quadratic is not constant. The slope-intercept formula will not work with parabolas.

What about (x – 3)2 ?

These are referred to as repeated solutions.

Find the solutions to this quadratic equation.

D) The solutions of a equation are also called the roots of the equation.

Parabolas: ...In Sports

Parabolas: ...In Archeticture

Parabolas: ...In Nature

Parabolas: ...Everywhere

And, of course, the most importantParabola of all

Solving Quadratic Equations by Factoring

Let's look at some of the different types of equations you'll face and how to deal with each of them

1: Set the equation = to 0 and solve:Example A. x2 + 6x + 9

x2 + 6x + 9 = 0; (x + 3) (x + 3) = 0, x = -3.This is a perfect square trinomial, and the parabola only touches the x axis at -3 and would be in this shape:

-3

Solving Quadratic Equations by Factoring

2. Solve x2 = 64. Remember the standard form?ax2 + bx + c = 0, where only a cannot = 0

In this case, b is 0, and c is 64.

We can solve by taking the square root of both sides.

3. Solve: 2x2 - x = 3Place all terms to the left of the = sign (Standard Form).

Since there is a leading coefficient (2), you willhave to divide at the end to obtain your solution.

The result is x = + 8; x = 8, and x = -8

4. Solve: x2 = 5x ***Do not cancel an 'x' from each side.

Factor GCF first, then solve

Applying Quadratic Equations:

Many different types of problems can be solved by use of a quadratic equation. Here are two.

1. The sum of an integer and its square is 72. What is the integer?

Applying Quadratic Equations:

(x2 + 20x) = 78 2

1. Find the base & height of the triangle below having an Area of 78 sq. yards

x2 + 20x = 156

x2 + 20x - 156 = 0

x = 6 or x = - 26 base = 6 yds., Height = 26 yds.

x2 + 20x - 156 = 0

Factor the trinomial

(x + )(x - )

Class Work:

Class Work 3.10 & Study Guide are due

tomorrow by end of class

Ask if you are not sure!!