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Foundations of Math 2

Unit 5: Solving Equations

Academics High School Mathematics

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5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties

On the graph below, graph the following two lines:

Y = 2x – 4

Y = 10

What is the point of intersection?

Explain what this point represents.

5.1 Solving Linear Equations Using Graphing, Tables, and Algebraic Properties Lesson

(Lesson from http://www.bigideasmath.com)

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5.1 Practice Solving Linear Equations Using Graphing, Tables, and Algebraic Properties

5.2 Warm Up – Solving Quadratic Equations by Graphing and Tables

Using a table of values, graph the function f(x)= x2 + 2x – 8

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Identify the following key characteristics:

Domain:

Range:

Vertex:

Axis of Symmetry:

x-intercept(s):

y-intercept:

Increasing Interval:

Decreasing Interval:

5.2 Lesson Handout - Solving Quadratic Equations by Graphing and Tables

Graphically Using a TableGraph the related quadratic Create a table for thefunction and find the x-intercepts quadratic function and find(because this is where y = 0). the x values for which y = 0.

Today we will be using quadratic functions to assist us in solving quadratic equations.

___________________________, where a, b, and c are real numbers and a ≠ 0 . The solutions of a quadratic equation are called its roots or zeros.

Definition: A quadratic equation is an equation that can be written in the standard form

Notice the similarity between a quadratic equation and a quadratic function. Quadratic Function: y = ax 2 + bx + c

Quadratic Equation: 0 = ax 2 + bx + c

SOLVING A QUADRATIC EQUATION BY USING ITS RELATED QUADRATIC FUNCTION

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x –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

y 20 9 0 –7 –12 –15 –16 –15 –12 –7 0 9 20

Exercise #1: The graph of the function y = x2 − 3x − 4 is given. According to the graph, the roots of ythe equation x2 − 3x − 4 = 0 are ______________________

Exercise #2: A table of values is given below for the function y = x2 − 2x −15. Use the table to determine the values of x for which x2 − 2x −15 = 0.

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Exercise #3: Find the zeros of the function y = x2 + 3x − 54 numerically by using a table of values. Create the table with your graphing calculator.

Exercise #4: The graph of a particular function of the form y = ax 2 + bx + c, where a, b, and c are real numbers, is shown below. Use the graph to answer the following questions.

(a) Is the numerical value of a positive or negative? Justify.

(b) State the numerical value of c. How do you know?

(c) State all solutions to each equation below:

(a) ax2 + bx + c = 0 has two (b) ax2 + bx + c = 0 has exactly (c) ax2 + bx + c = 0 has roots; a > 0 one root; a < 0 no roots; a > 0

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Fact: A quadratic equation can have _____, ______, or ______ real solutions

Fact: A quadratic equation can have _____, ______, or ______ real solutions

(i) ax 2 + bx + c = 0. (ii) ax 2 + bx + c = 3.

Exercise #5: Using the accompanying grids, sketch graphs of functions of the form y = ax 2 + bx + c that satisfy the given criteria.

Exercise #6: Determine the roots for each quadratic equation given below by using your graphing calculator.

(a) x2 + x − 6 = 0 (b) − x2 − 6x − 9 = 0 (c) x2 + 4x + 6 = 0

Exercise #7: Which of the following functions has two real zeros?

(1) y = x2 + 7x +14 (3) y =− x2 − 5x + 6 (2) y = x2 −10 x + 25 (4) y = x2 + 4

Exercise #8: If one x-intercept of the graph of a quadratic function is −4 and the axis of symmetry has equation x = 3, then what is the other x-intercept?

Exercise #9: Which table below illustrates a quadratic function with a maximum value of zero?

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4. x2 + 2x – 3 = 05. –m2 + 8m – 16 = 06. –g2 + 4g – 5 = 0

7. 4k2 – 8k = -48. h2 – 3 = 0 9. n2 + 6 = 4n

10. w2 = -2w 11. –v2 = -6v + 7 12. t2 = 4

13. The real roots of a quadratic equation correspond to the __? of the graph of the related function.

A. x-interceptsB. y-interceptsC. vertex D. maximum

5.2 Homework - Solving Quadratic Equations by Graphing and Tables

5.2 Practice – Solving Quadratic Equations by Graphing and Tables

Solving Quadratic Equations by GraphingThe solutions of a quadratic equation are called the roots of the

equation. You can find the real number roots by finding the x-intercepts or zeros of the related quadratic function. Quadratic equations can have two distinct real roots, one distinct root, or no real roots. These roots can be found by graphing the equation to see where the parabola crosses the x-axis.

a) b) c)

State the real roots of each quadratic equation whose related function is graphed below. 1.

2. 3.

Solve each equation by graphing. If integral roots cannot be found, state the consecutive integers between which the roots lie.

Describe the real roots of the quadratic equations whose related functions are graphed below.

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1. The graph of y = x2 − 6x + 8 is shown. The roots of the equation x2 − 6x + 8 = 0 are __________________

2. Which of the following graphs illustrates a quadratic function that has no real zeros?

(A) (B)

(C) (D)

(a) y = x 2+ 3x −10 (b) y = − x2 − 5x − 6 (c) y = x2 + 8x +16

(d) y = x2 + 2 (e) y = − x2 − 2x − 4 (f) y = − 25 −10 x – x2

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5.3 Warm- Up Greatest Common

Factor

Find the greatest common factor for each of the following:

1. 12,48

x –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4

y –13 0 11 20 27 32 35 36 35 32 27 20 11 0 –13

(4)

x

3. Determine the zeros for each quadratic function given below by using your graphing calculator.

4. A table of values is given below for the function y = 27 − 6x − x2. Use the table to determine the values of x for which 27 − 6x − x2 = 0.

5. How many real roots does x2 + 7x + 6 = 0 have?

6. If the two x-intercepts of the graph of a quadratic function are −3 and 9, then the equation of the axis of symmetry is

7. If one x-intercept of the graph of a quadratic function is 4 and the axis of symmetry has an equation of x = 7, then what is the other x-intercept?

8. The graph of a particular function of the form y = ax 2 + bx + c, where a, b, and c are real numbers, is shown below. Use the graph to solve each of the following equations.

(a) ax2 + bx + c = 0 (b) ax2 + bx + c = 1 (c) ax 2+ bx + c = − 3

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2. 35 m2n , 21m3 n

3. 80 x3 ,30 yx2

4. 140 n❑ ,140 m2 , 80 m2

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5.3 Lesson Handout – Solving Quadratic Equations by Factoring

Review of Factoring Using the “X-Box” Method

Example: 3n2-6n-45

STEPS: Example:

1. Factor out GCF. Complete the remaining steps on the remaining trinomial.

2. Draw a big X.

3. Multiply the lead coefficient and the constant together. Put the answer in the top of the X.

4. Put the middle term number of the trinomial in the bottom of the X.

5. Find two numbers that will

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multiply to give you the top number and add to give you the bottom number.

6. Put those two numbers in the sides of the X.

7. Rewrite the original trinomial by removing the middle term and replacing the term as a sum by using the two numbers you found in step 5. (Don’t forget the variables!)

8. Put the polynomial from step 6 into a box.

9. Factor the GCF from each row and each column of the box. Place

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these numbers on the top and the side of the box.

10. Read your answer from the sides of the box. (Don’t forget the GCF!)

Now try these:

1. f(x) = 12x2+17x+6 answer:_______________________

2. f(x) = 4x2-100y2 answer:_______________________

3. f(x) = x2+14x+24 answer:_______________________

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4. f(x) = 6x3+15x2-9x answer:_______________________

5. f(x) = 2h2-3h-18 answer:_______________________

6. f(x) = a2+18a+81 answer:_______________________

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7. f(x) = 32x2-80x+50 answer:_______________________

8. f(x) = 6y2-5y-6 answer:_______________________

Solving Quadratics by Factoring

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Steps: Example:

1. Quadratic Equation:

n2 = -18 – 9n

2. Put equation into standard form and set equal to zero:

3. Factor the equation:

4. Set each factor equal to zero:

5. Solve each smaller equation:

Now try these: Solve each by factoring

a. (x + 3)(x – 7) = 0 b.

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c. d.

e. f.

5.3 Practice – Solving Quadratic Equations by Factoring

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5.3 Homework – Solving Quadratic Equations by Factoring

Solve each quadratic by factoring.

1. x(x-7) = 0 2. p2 - 19p + 70 = 0

3. (2z + 1)2 = 0 4. y2 = 3y

5. c2 - 9 = 0 6. h2 + 14 = -9h

7. -f - 6 = -f2 8. 23x - 6 = -4x2

9. r3 - 2r2 - 15r = 0 10. z2 - z = 30

11. 4v2 + 3v = 10 12. x2 + 15x = -44

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5.4 Warm-Up Factoring

Factor completely:

1. x2−7 x−18

2. 7 k2+9 k

3. 7 x2−45 x−28

Explain how you know to just factor or to solve a quadractic.

Explain the steps for factoring a quadratic.

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5.4 Lesson Handout – Solving Quadratics Using the Quadratic Formula

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5.4 Practice – Solving Quadratics Using the Quadratic Formula

Solve each equation using the quadratic formula:x=−b±√b2−4 ac

2 a

1. x2 – 7x + 10 = 0 2. x2 – 14x + 45 = 0

3. 0.25x2 – 0.25x – 10.5 = 0 4. 0.3y2 + 0.15y – 11.7 = 0

5. 10k2 + 200k + 937.5 = 0 ` 6. x2 – 11x + 30.25

7. 4x2 + 8x – 77 8. 16y2 – 8y – 3 = 0

9. x2 – 1.375x + 0.375 = 0 10. z2 + 0.6z – 20.16 = 0

11. x2 = 15 + 2x 12. x2 – 10x + 35 = 7x – 35

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5.4 Homework – Solving Quadratics Using the Quadratic Formula

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5.5 Warm Up

EXPLORING MULTIPLYING POLYNOMIALS

x2 = –x2 = x = –x = +1 = –1 =

ex. 1. x(x + 1) 2. 2x(–x + 3)

X(x+1) = x2 + x

2x(-x + 3) = -2x2 + 6x

Try these examples:

1. 2(3x + 1) 2. x(x + 2)

3. –2x(x + 2) 4. x(2x – 1)

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EXPLORING MULTIPLYING BINOMIALS

ex. 1. (x + 1)2 2. (x + 1)(–x + 3)

(x + 1)(x + 1) = x2 + 2x + 1

(x + 1)(–x + 3) = -x2 + 2x + 3

Try these examples:

1. (x + 2)(x + 1) 2. (x + 2)2

3. (–2x – 1)(x + 2) 4. (x –3)(x - 2)

x2 -x -x -x

- x- x- x

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5.5 Completing the Square with Algebra Tiles

Completing the Square with Algebra Tiles Name

We have used square roots and factoring to solve quadratic equations. The technique we will learn today is called “completing the square.” Completing the square is a useful method to solve any type of quadratic equations, especially those that cannot be factored.

In this activity we will be completing the square for expressions written in the form x2 + bx. The process of completing the square will allow us to write x2 + bx as a binomial square and as a perfect square trinomial.

We want to complete the square of x2 6x , so first let’s model this with algebra tiles. Remember, when

working with algebra tiles, the big square represents the quadratic term x 2 , the little rectangles represent the linear x term, and the tiny squares represent the constant term. The red side represents negative terms for all of these tiles, and the blue, green, and yellow sides represent positive terms.

Figure # 1

The x 2 term will always be in the top left corner of your diagram.Our next step is to align the linear x terms evenly along the big square. We can

do this like the diagram shown. Use the tiles at your desk and create the configuration as in figure #1.

Notice that the bottom right hand corner, in figure #1, is missing and this where ‘completing the square’ comes in. We will fill in that empty space with 9 tiny squares to turn the diagram into figure #2.

Figure # 2 Notice the dimensions of our newly completed square. The square is (x 3) on Completed Square one side, and (x 3) for the other side. The area of this completed square is (x – 3)(x-3) or (x – 3)2 written as a binomial square. This is the same as x2 – 6x + 9 written as a perfect square trinomial.

Therefore, completing the square for x2 – 6xleads us to obtain the perfect square trinomial x2 – 6x + 9 which can also be written as the binomial square (x – 3)2

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5.5 Group ActivityUsing your group’s collection of algebra tiles, create a model for each of the expressions and use your findings to fill in the chart.

Expression

x2 + bx Model

Number of tiny 1-tiles needed

to complete the square

Sketch of the SquareExpression written as

a perfect square

trinomialx2 + bx + c

Expression written

as abinomial square

x2 – 6x Look at figure # 2

x2 2x

x2 4x

x2 8x

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Explore how to complete a square when given a value for “c”

Standard Form

Number of x2

TilesNumber of x Tiles

Number of Unit Tiles

Sketch of the SquareTiles

added to complete

the square (Add

only zero pairs!)

Length of the

SquareArea of

the Square(Length)2

Unit TilesLeft Over

(Don’t forget to

take out the zero pairs.)

Expression Combining

Previous Two Columns

EX:x2 – 2x + 3 1 -2 3 0 x-1 (x-1)2 +2 (x-1)2 + 2

EX:x2+ 6x -3 1 6 -3

9 (to complete the square)

And

9 (to make the zero pairs)

x+3 (x+3)2 -3 + -9 = -12

(x+3)2 - 12

EX:x2-4x+1 1 -4 1

3 (to complete the square)

And

3 (to make the zero pairs)

x-2 (x-2)2-3 (x-2)2 - 3

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Now you try:

Standard Form Number of x2

Tiles

Number of x Tiles

Number of Unit Tiles

Sketch of the Square

Tiles added

to complet

e the square (Add only zero

pairs!)

Length of the

Square

Area of the

Square(Length)2

Unit TilesLeft Over

(Don’t forget to take out the zero pairs.)

Expression Combining Previous

Two Columns

x2 –2x + 4

x2 + 2x - 5

x2 - 8x + 3

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5.5 Practice

Standard Form Number of x2

Tiles

Number of x Tiles

Number of Unit Tiles

Sketch of the Square

Tiles added to complet

e the square (Add only zero

pairs!)

Length of the

Square

Area of the

Square(Length)2

Unit TilesLeft Over

(Don’t forget to take out the zero pairs.)

Expression Combining Previous

Two Columns

x2 + 2x - 3

x2 + 4x + 1

x2 - 6x + 8

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5.5 Homework: Completing the Square

Standard Form Number of x2

Tiles

Number of x Tiles

Number of Unit Tiles

Sketch of the Square

Tiles added to complet

e the square (Add only zero

pairs!)

Length of the

Square

Area of the

Square(Length)2

Unit TilesLeft Over

(Don’t forget to take out the zero pairs.)

Expression Combining Previous

Two Columns

x2 + 4x - 5

x2 + 6x + 2

x2 - 8x - 8

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5.6 Warm-Up Solving Equations

Solve. Explain your reasoning behind why you solved each problem the way you did.

1. (4 k+5 ) (k+1 )=0

2. n2−10 n+22=−2

3. 7 r2−14 r=−7

4. 5 r2−44 r+120=−30+11r

5. 15 a2−3a=3−7 a

5.6 Practice-Completing the Square

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5.6 Homework: Completing the Square

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5.7 Warm-Up Graphing Radical Functions

Graph each function using your calculator. Sketch a picture of what you see. Use a standard viewing window. Make sure you plot key points on your sketch (y-intercept and x-intercept(s))

1. y=√x−2+5

2. y=√x+5

3. y=−2√ x+2

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5.7 Practice-Solving Radical Functions By Graphing and Algebraic Properties

5.7 Homework-Solving Radical Functions By Graphing and Algebraic Properties

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5.8 Warm-Up Adding and Subtracting Fractions and Solving Proportions

Add or subtract the following fractions.

1. 1. 35+ 4

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2. 2a+7

2

3. 34−2

3+ 1

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Solve each proportion.

4. 2x+1

= 5x−3

5. 3x−1

= x+16

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5.8 Practice- Solving Rational Equations

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5.8 Homework – Solving Rational Equations

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