H – Quadratics, Lesson 1, Solving Quadratics (r. 2018) - JMAP

H ? Quadratics, Lesson 1, Solving Quadratics (r. 2018)

QUADRATICS Solving Quadratics

Common Core Standards

A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Next Generation Standards

AI-A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Shared standard with Algebra II)

A-REI.B.4a Solve quadratic equations in one varia- AI-A.REI.4 Solve quadratic equations in one variable.

ble.

Note: Solutions may include simplifying radicals.

NYSED: Solutions may include simplifying radicals.

NOTE: This lesson is in four parts and typically requires four or more days to complete.

LEARNING OBJECTIVES

Students will be able to:

1) Transform a quadratic equation into standard form and identify the values of a, b, and c. 2) Convert factors of quadratics to solutions. 3) Convert solutions of quadratics to factors. 4) Solve quadratics using the quadratic formula. 5) Solve quadratics using the completing the square method. 6) Solve quadratics using the factoring by grouping method.

Teacher Centered Introduction

Overview of Lesson - activate students' prior knowledge - vocabulary - learning objective(s) - big ideas: direct instruction - modeling

Overview of Lesson Student Centered Activities

guided practice Teacher: anticipates, monitors, selects, sequences, and connects student work

- developing essential skills

- Regents exam questions

- formative assessment assignment (exit slip, explain the math, or journal entry)

box method of factoring completing the square constant factoring by grouping factors forms of a quadratic linear term multiplication property of zero

VOCABULARY quadratic equation quadratic formula quadratic term roots solutions standard form of a quadratic x-axis intercepts zeros

Part 1 ? Overview of Quadratics

BIG IDEAS

The standard form of a quadratic is: ax2 + bx + c =0 . ? ax2 is the quadratic term ? bx is the linear term ? c is the constant term

Note: If the quadratic terms is removed, the remaining terms are a linear equation.

The definition of a quadratic equation is: an equation of the second degree.

Examples of quadratics in different forms:

Forms

Examples

standard form

6x? + 11x ? 35 = 0

2x? ? 4x ? 2 = 0

-4x? ? 7x +12 = 0

20x? ?15x ? 10 = 0

x? ? x ? 3 = 0

5x? ? 2x ? 9 = 0

3x? + 4x + 2 = 0

without the bx term (the linear term)

-x? + 6x + 18 =0 2x? ? 64 = 0

x? ? 16 = 0

9x? + 49 = 0

-2x? ? 4 = 0

4x? + 81 = 0

-x? ? 9 = 0

3x? ? 36 = 0

without the c term (the constant term )

6x? + 144 = 0 x? ? 7x = 0

2x? + 8x = 0

-x? ? 9x = 0

x? + 2x = 0

-6x? ? 3x = 0

-5x? + x =0

-12x? + 13x = 0

11x? - 27x = 0

factored forms

( x + 2)( x ? 3) = 0

( x + 1)( x + 6) = 0

( x ? 6)( x + 1) = 0

( x - 5)( x + 3) = 0

( x - 5)( x + 2) = 0

( x - 4)( x + 2) = 0

(2x + 3)(3x - 2) = 0

?3( x ? 4)(2x + 3) = 0

other forms

x(x ? 2) = 4

x (2x + 3) = 12

3x ( x + 8) =- 2

5x=? 9 - x

-6x? = - 2 + x

= x? 27x -14

x? + 2x = 1

4x? - 7x = 15

-8x? + 3x =-100

25x + 6 = 99 x? (Source: your )

Multiplication Property of Zero: The multiplication property of zero says that if the product

of two numbers or expressions is zero, then one or both of the numbers or expressions must equal

zero. More simply, if

, then either

or , or, both x and y equal zero.

Example: The quadratic equation ( x + 2) (x - 4) = 0 has two factors: ( x + 2) and (x - 4) . The

multiplication property of zero says that one or both of these factors must equal zero, because the

product of these two factors is zero. Therefore, write two equations, as follows:

Eq #1

( x + 2) =0 Therefore, x = -2

Eq #2

(x - 4) = 0 Therefore, x = 4

By the multiplication property of zero, x ={-2, +4} .

Zeros: A zero of a quadratic equation is a solution or root of the equation. The words zero, solution, and root all mean the same thing. The zeros of a quadratic equation are the value(s) of x when . A quadratic equation can have one, two, or no zeros. There are four general strategies for finding the zeros of a quadratic equation:

1) Solve the quadratic equation using the quadratic formula. 2) Solve the quadratic equation using the completing the square method. 3) Solve the quadratic equation using the factoring by grouping method. 4) Input the quadratic equation into a graphing calculator and find the x-axis intercepts.

x-axis intercepts: The zeros of a quadratic can be found by inspecting the graph view of the equation. In graph form, the zeros of a quadratic equation are the x-values of the coordinates of the x-axis intercepts of the graph of the equation. The graph of a quadratic equation is called a parabola and can intercept the x-axis in one, two, or no places.

Example: Find the x-axis intercepts of the quadratic equation ( x + 2) (x - 4) = 0 by inspecting

the x-axis intercepts of its graph.

The coordinates of the x-axis intercepts are are (-2, 0) and (4, 0) . These x-axis intercepts show

that the values of x when y=0 are -2 and 4, so the solutions of the quadratic equation are

x ={-2, +4} .

The Difference Between Zeros and Factors Factor: A factor is:

1) a whole number that is a divisor of another number, or 2) an algebraic expression that is a divisor of another algebraic expression. Examples:

o 1, 2, 3, 4, 6, and 12 all divide the number 12, so 1, 2, 3, 4, 6, and 12 are all factors of 12.

o ( x - 3) and ( x + 2) will divide the trinomial expression x2 - x - 6 , so ( x - 3) and ( x + 2) are both factors of the x2 - x - 6 .

Start with Factors and Find Zeros Remember that the factors of an expression are related to the zeros of the expression by the multiplication property of zero. Thus, if you know the factors, it is easy to find the zeros.

Example: If the factors of the quadratic equation 2x2 + 5x + 6 =0 are (2x + 2) and ( x + 3) , then by the multiplication property of zero: either (2x + 2) = 0 , or ( x + 3) = 0 , or both equal

zero. Solving each equation for x results in the zeros of the equation, as follows:

(2x + 2) = 0

2x = -2

( x + 3) = 0

x = -1

x = -3

Start with Zeros and Find Factors

If you know the zeros of an expression, you can work backwards using the multiplication

property of zero to find the factors of the expression. For example, if you inspect the graph of

an equation and find that it has x-intercepts at (3, 0) and (-2, 0) , then you know that the

solutions are x = 3 and x = -2 . You can use these two equations to find the factors of the quadratic expression, as follows:

x=3

( x - 3) =0

x = -2

( x + 2) = 0 The factors of a quadratic equation with zeros of 3 and -2 are ( x - 3) and ( x + 2) .

With practice, you can probably move back and forth between the zeros of an expression and the factors of an expression with ease.

Part 1 ? Overview of Quadratics

DEVELOPING ESSENTIAL SKILLS

Convert the following quadratic equations to standard form and identify the values of a, b, and c:

x(x ? 2) = 4

x2 - 2x - 4 =0 a= 1 , b=-2 , c=-4

x (2x + 3) = 12

2x2 + 6x -12 = 0 a= 2, b=6 , c= -12

3x ( x + 8) =- 2

3x2 + 24x + 2 =0 a= 3 , b= 24, c= 2

5x=? 9 - x

= 5x? + x - 9 0 a= 5 , b= 1, c= -9

-6x? = - 2 + x

-6x? - x + 2 =0 a= -6, b= -1, c= 2

= x? 27x -14

x? - 27x +14 = 0 a= 1 , b=-27 , c= 14

x? + 2x = 1

x? + 2x -1= 0 a= 1 , b=2 , c=-1

4x? - 7x = 15

4x? - 7x -15 = 0 a= 4 , b=-7 , c=-1

-8x? + 3x =-100

-8x? + 3x +100 = 0 a= -8 , b=3 , c=100

25x + 6 = 99 x?

-99 x? + 25x + 6 = 0 a= -99 , b=25 , c=6

2x? = 64

2x? ? 64 = 0 a=2 , b= 0, c=

0 =-16 + x?

x? ? 16 = 0 a=1 , b= 0, c= -16

49 = - 9x?

9x? + 49 = 0 a= 9 , b=0 , c=49

x? = 7x

x? ? 7x = 0 a= 1 , b=-7 , c=0

2x? = - + 8x

2x? + 8x = 0 a= 2 , b=8 , c= 0

0 =-9x - x?

-x? ? 9x = 0 a=-1 , b=-9 , c= 0

Find the zeros of the following quadratic equations:

a. x = {-2,3} a. ( x + 2)( x ? 3) = 0 b. x = {-6,1}

b. ( x + 1)( x + 6) = 0 c. ( x ? 6)( x + 1) = 0 d. ( x - 5)( x + 3) = 0

c. x = {-1, 6} d. x = {3,5} e. x = {2,5}

e. ( x - 5)( x + 2) = 0 f . ( x - 4)( x + 2) = 0 g. (2x + 3)(3x - 2) = 0

f . x = {-2, 4}

g.

x=

-

3 2

,

2 3

h. ? 3( x ? 4)(2x + 3) = 0

h.

x=

-

3 2

,

4

Part 2 ? The Quadratic Formula

The quadratic formula is: x = -b ? b2 - 4ac 2a

Quadratic Formula Song

SOLVING QUADRATIC EQUATIONS STRATEGY #1: Use the Quadratic Formula

Start with any quadratic equation in the

form of

The right expression must be zero.

Identify the values of a, b, and c.

,

, and

Substitute the values of a, b, and c into

the quadratic formula, which is

Solve for x

The quadratic formula can be used to solve any quadratic equation.

Part 2 ? The Quadratic Formula

DEVELOPING ESSENTIAL SKILLS

Solve the following quadratic equations using the quadratic formula. Leave answers in simplest

radical form.

x? ? x ? 3 = 0

x = -b ? b2 - 4ac 2a

a = 1, b = -1, c = -3

-(-1) ? (-1)2 - 4(1)(-3)

x=

2 (1)

x = 1? 1+12 2

x = 1? 13 2

20x? ?15x ? 10 = 0 2x? ? 4x ? 2 = 0

x = -b ? b2 - 4ac 2a

a = 20, b = -15, c = -10

-(-15) ? (-15)2 - 4(20)(-10)

x=

2 ( 20 )

x = 15 ? 225 + 800 40

x = 15 ? 1025 40

x = 15 ? 25 ? 41 40

x = 15 ? 5 41 40

x = 3 ? 41 8

x = -b ? b2 - 4ac 2a

a=2, b=-4,c=-2

-(-4) ? (-4)2 - 4(2)(-2)

x=

2(2)

x = 4 ? 16 +16 4

x = 4 ? 32 4

x = 4 ? 16 ? 2 4

x= 4?4 2 4

x= 1? 2

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