Hampton math · Web viewI. Simplifying Radicals, Solving Quadratics by Square Roots, and Complex...

Common Core Math 2: Unit 2B – Solving Quadratic Functions

I. Simplifying Radicals, Solving Quadratics by Square Roots, and Complex Numbers

Real Numbers: Rational Numbers: Irrational Numbers:

Part 1: Simplifying Radicals

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

Reverse the process. Un-simplify the simplified radical form. Write as one number under the radicand.

11. 12.

13. 14.

Simplify.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

Operations with Rational and Irrational Numbers

Complete the addition table.

+ 5 05 10 5.5

0

Complete the multiplication table.

5 05 25 2.5

0

Based on the above information, conjecture which of the statements is ALWAYS true, which is SOMETIMES true, and which is NEVER true?25. The sum of a rational number and a rational number is rational.

26. The sum of a rational number and an irrational number is irrational.

27. The sum of an irrational number and an irrational number is irrational.

28. The product of a rational number and a rational number is rational.

29. The product of a rational number and an irrational number is irrational.

30. The product of an irrational number and an irrational number is irrational.

Solving Quadratics by Square Roots

Some quadratic equations – ax2 + bx + c = 0 – can be solved by taking the square root. This method works if ________________________________________________________.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Imaginary Numbers:

Examples:

Complex Numbers:

Examples:

Solve the following quadratic equation by taking the square root.

This equation has ___________________________________________________, but you can state solutions

using ________________________________ and ____________________________ numbers.

Solve and simplify completely. 41. 42.

43. 44.

Practice:Simplify completely.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

Reverse the process. Un-simplify the simplified radical form. Write as one number under the radicand.

13. 14.

15. 16.

Simplify completely.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

Solving each quadratic equation by taking the square root. 27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

II. Solving Quadratics by Factoring

1. Put equation in _________________________________. (ax2 + bx + c = 0) 2. ____________________________ completely. 3. Set each factor equal to __________________. 4. Solve each _______________________________.

Solve each equation by factoring. 1. (x – 7)(x + 3) = 0 2. (3x – 2)(4x + 1) = 0

3. m(m – 3) = 0 4. 4(x + 3)(x + 9) = 0

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

Practice: Solve these equations by factoring.1. (3x + 2)(2x + 5) = 0 2. 8(5x – 1)(x + 1) = 0

3. k(2k – 7) = 0 4. n2 + 9n + 18 = 0

5. 3x2 – 33x + 72 = 0 6. 7x2 + 35x – 42 = 0

7. x2 = -4x – 4 8. -2x2 – x + 12 = -3x2 + 6x

9. -4x2 + 6x – 16 = -5x2 10. 8x2 + 3x + 2 = 7x2

11. x2 + x = 2 12. 10x2 – 35 = 65x

13. 3x2 – 8x = 16 14. 16x2 – 114x = -14

15. 28x2 = -96 – 184x 16. 7x2 + 32 = 7 – 40x

Solve each equation by factoring or by taking square roots. Simplify each solution completely.

III. Solving Quadratics by Completing the SquareIf the Coefficient of x

2=1

1. Move the _______________ term to the _______________ side of the equation.2. Add ____________to both sides of the equation.3. _____________ the perfect square trinomial on the left and simplify the right.4. Take the square root of both sides (remember the__________)5. Write both equations separately and _______________ each.

Solve each equation by completing the square. 1. x2 – 20x + 32 = 0 2. x2 + 9x = 136

3. m2 – 6m = 247 4. x2 + 5x + 3 = 0

5. x2 – 14x + 16 = 0 6. x2 – 7x = 0

7. a2 + 12a + 30 = 0 8. x2 – 4x = -16

9. b2 + 12b + 20 = 0 10.

11. n2 +18n + 80 = 0 12.

13. 14.

15. x2 +10x – 20 = 0 16. x2 – 22x + 130 = 0

Khan Academy: Need extra practice with completing the square where a is not equal to 1? Use the QR code to watch!

If the Coefficient of x2≠11. Move the _______________ term to the _______________ side of the equation.2. Divide a from ax2+bx.3. Complete the square by adding ______________ to a¿4. Add ______________ to the right side of the equation.5. _____________ the perfect square trinomial on the left and simplify the right.6. Divide by a.7. Take the square root of both sides (remember the__________)8. Write both equations separately and _______________ each.

Solve each equation by completing the square.1. 2x2 + 8x – 5 = 0 2. 8x = 4x2 – 1

3. 2x2 – 4x + 5 = 6 4. 2x2 – 12x + 7 = 0

5. 3x2 + 6x – 5 = 0 6. -2x2 + 16x +1 = 0

7. 5k2 = 60 – 20k 8. 6x2 – 48 = -12x

9. 8a2 + 16a = 42 10. 9n2 + 79 = -18n

11. 2n2 = -6 + 8a 12. 4n2 + 4n + 36 = 0

13. 5n2 – 20n = -6 14. 3a2 + 6a – 34 = 0

15. 2v2 + 5v – 7 = 0 16. 4n2 + 11n = 15

IV. Solving Quadratics with the Quadratic Formula

The quadratic formula is used to find _________________________ of a quadratic equation in the form: y = ax2 + bx + c.

Quadratic Formula: The quadratic formula can always / sometimes / never be used to solve a quadratic equation.

It is best to use the quadratic formula when the quadratic equation is not ____________________.

Practice: Use the quadratic formula by using it to solve the following equations, then graph the quadratic functions to see how the solutions appear as x-intercepts.

1. 5x2 + 3x – 2 = 0 2. 10 – x2 – 3x = 0 a = ________ b = ________ c = ________ a = ________ b = ________ c = ________

3. What information about the graph is provided by and ?

4. What information about the graph is provided by the expression ?

5. What information about the graph is provided by the expression ?

Practice: Use the quadratic formula to find the solutions of the quadratic functions. Report your answers in exact form, using radicals where necessary rather than decimal approximations.

The Quadratic Formula can also be written as :

6. x2 = x + 3 7. 2x2 + 23 = 14x

8. x2 = 2x + 48 9. 2x2 + 29 = -18x

10. 5x2 + 3x - 3 = 0 11. 5x2 + 50x = -125

12. Use the quadratic formula to check each of the following claims about equations and solutions.a. The solutions of x2 + 5x – 6 = 0 are integers 6 and -1.

b. The solutions of 6x2 + x – 2 = 0 are rational numbers and .

c. The solutions of x2 – 6x + 4 = 0 are irrational

numbers and .d. The equation x2 + 5x + 7 = 0 has no real solutions.

Complex Numbers

When trying to solve x2 – 6x + 13 = 0, the quadratic formula gives

We have already discovered that when discriminant is negative, then the quadratic equation has no real number solutions.

13. Sketch a graph of the function f(x) = x2 – 6x + 13 = 0. Then, explain how the graph shows that there are no real number solutions for the equation.

14. Write the solutions to the quadratic equation using complex numbers.

15. Use the quadratic formula to show that each of these equations has complex number solutions with nonzero imaginary parts. Express those solutions in the form a + bi where a and b are real numbers.a. x2 – 10x + 29 b. 4x2 + 16 = 0

c. x2 – 4x + 13 = 0 d. 3x2 – 18x + 30 = 0

V. Comparing methods for Solving Quadratic FunctionsThere are several different methods that can be used to solve quadratic equations. Factoring, completing the square and then taking the square root, quadratic formula, using tables, and using graphs are all great ways to solve quadratics. Does one method work better than another method when solving quadratic equations?

1. How can you solve quadratic equations like ax2 + b = c using algebraic reasoning?

2. How can you solve equations like ax2 + bx = 0 using algebraic reasoning?

3. How can you solve equations like ax2 + bx + c = 0 when the expression ax2 + bx + c can be written in equivalent form as the product of two linear expressions?

4. In what situations does it make sense to use the quadratic formula to solve an equation?

How could you use calculator to find solutions for a quadratic equation like 3x2 +7x – 12 = 0?

Match each of the following equations to the solution method that you think is the best for solving the equation. Provide a simple explanation for your choice. You can only use each method once.A- Taking the Square Root B- Factoring C- Completing the Square D- Using the Quadratic Formula

______1. 2x2 – 13x – 8 = 0

______2. 3x2 - 27 = 0

______3. x2 – 6x – 7 = 0

______4. x2 - 11x + 30 = 0

VI. The DiscriminantWarm UpUse the quadratic formula to solve each of the following.1. x2+10x+25=0 2. 2 x2−6 x+10=0 3. 3 x2−4 x−5=0 4. x2−8 x+15=0

1. What part of the quadratic formula calculations shows whether there will be 2, 1, or 0 real number solutions? How is that information revealed by the calculations?

2. What part of the quadratic formula calculations shows whether the solutions will be rational or irrational numbers? How is that information revealed by the calculations?

Discriminant:

Find the discriminant, then check the appropriate column for each equation.Equation Discriminant One

RationalRoot

Two Rational

Roots

Two Irrational

Roots

NOReal

Roots1. x2−6 x+11=2

2. 3 x2+5 x=12

3. 3 x2+48=0

4. x2−27=0

5. x2+ x+1=0

6. x2+4 x−1=0

PracticeFind the discriminant, then check the appropriate column for each equation.

EQUATION DISCRIMINANT ONE RATIONAL ROOT

TWO RATIONAL ROOTS

TWO IRRATIONAL ROOTS

NO REAL ROOTS

1. x2+x−2=02. 3 x2−22 x+7=03. x2−8 x+4=04. 2 x2−9 x+13=05. 4 x2−12 x−39=06. 4 x2−12 x+9=07. 4 x2−20 x−5=08. 15 x2+11 x−12=09. 5 x2−7 x+3=010.

6 x2−11x−21=0

11.

4 x2+20 x+25=0

12.

9 x2+30 x+25=0

13.

2 x2+6 x−9=0

14.

4 x2−12 x−19=0

15.

6 x2+10 x−5=0

16.

16 x2−8x+14=0

VII. Systems of Equations with Linear and Quadratic FunctionsWarm UpSolve the following system of linear equations by graphing.1. 2.

3. Solve the following systems by substitution.

4 x+2 y=10x− y=13

INVESTIGATION: MAKING MORE BY CHARGING LESS In most buisnesses, one of the most important tasks is setting prices for the goods or services that are bring offered for sale. For example, consider the case of producers who have a contract to bring a musical production to a summer theater.

They have to estimate costs of putting on the show, income from ticket sales and concessions, and the profit that can be made. Values of these variables depend on the number of tickets sold and the prices charged for tickets.

1. Data from a market survey suggest the following relationship between ticket price and number of tickets sold.

a. After graphing the data and experiment to find a function model for the pattern, the business planners proposed the function s (p )=2,500−50 p for this demand pattern. i. What do p and s(p) represent in that function? ii. Is that function reasonable?

Can you produce a better model?

b. What do the numbers 2,500 and -50 tell about the way ticket sales depend on ticket price?

2. Based on the relationship between ticket price and number of tickets sold, the show planners figured that income could be predicted from ticket price, as well. They reasoned that since income is equal to the product of price per ticket and number of tickets sold, I ( p )=p (2,500−50 p).a. Test this function rule by calculating the predicted income from ticket sales in two ways.

i. First, use the data in Problem 1 to estimate income if the ticket price is set at $10, $20, $40.

ii. Then use the function to calculate predicted ticket income for the same ticket prices.

b. Sarren proposed the function I ( p )=2,500 p−50 p2 for predicting income from ticket price. Is this equivalent to I ( p )=p (2,500−50 p)? Why or why not?

3. The next step in making business plans for the production was to estimate operating costs. Some costs were fixed (for example, pay for the cast and rent of the theater), but other costs would depend on the number of tickets sold s (for example, number of ushers and ticket takers needed). After estimating all of the possible operating costs, the function c (s )=17,500+2 s was proposed.

a. According to the rule, what are the fixed operating costs and the costs per customer?

b. Daniel proposed the rule c ( p )=17,500+2(2,500−50 p). Is this correct? Why or why not?

c. Minta suggested the expression 17,500+2(2,500−50 p) in part b could be simplified to 22,500−100 p . Is this correct? Why or why not?

4. The following graph shows how income and operating cost depend on ticket price and how they are related to each other.

a. Use the graph to estimate answers for the following questions, and explain how you arrive at each estimate.

i. For what ticket price(s) will operating cost exceed income?

ii. For what ticket price(s) will income exceed operating cost?

iii. For what ticket price(s) will income equal operating cost?

b. Write an equation that helps in locating the break-even point(s)-the ticket prices for which income exactly equals operating cost.

5. It is likely that the show producers want to do more than break-even. They will probably seek maximum profit.a. Use the income and operating cost functions to write a function showing how profit depends on ticket price. Write the profit function F(p) in two equivalent forms – one that shows the expressions for income and cost and the other that is the simples for calculating the profit.

b. Use the profit function to estimate the maximum profit plan – the ticket prices that will lead to maximum profit and the dollar profit that will be made at that price.

c. Use the results from Part b to calculate the number of tickets sold and the operating cost in the maximum profit situation.

6. Explore equations in the form mx+d=a x2+bx+ca. Sketch function graphs that illustrate the possible number of solutions for equations involving linear and quadratic functions.

b. Solve each of the following equations algebraically.i. x2+4 x−5=x+5 ii. x2−2 x+3=−2 x+3 iii. −3 x2−6 x+1=−4 x+6

c. Sketch graphs of the linear and quadratic functions involved in part b and explain how the graphs illustrate the solutions.

i. ii. iii.

Practice: Solve each of these equations, sketch graphs showing the functions involved, and label points corresponding to solutions with their coordinates.1. x+2=x2−4 x+6 2. −x2+2x+3=2x−1

3. 2 x+4=4−x2 4. x2−2 x=−x+6

5. Solve the following system algebraically: 6. Graphically solve the system of equations in the problem to the left.

7. Which graph could be used to find the solution to the following system of equations?

(1) (2) (3) (4)

VIII. Quadratic InequalitiesWarm Up

Use symbolic reasoning to find all solutions for these equations. Illustrate each solution by a sketch of the graphs of the functions involved, labeling key points with their coordinates.

a. 2 x2−4 x=4−2 x b. −3−x=x2−6 x+1

INVESTIGATION: QUADRATIC INEQUALITIES (Adapted from Core Plus Course 3 pg. 112)

1. Consider the inequality t 2−t−6≤0.a. Which of these diagrams is most like what you would expect for a graph of the functiong ( t )=t 2−t−6?

How can you decide without using a graphing tool?

b. The expression t 2−t−6 can be written in equivalent factored from as ( t−3 ) ( t+2 ) . How can this fact be used to solve the equation t 2−t−6=0? What do those solutions tell about the graph of g(t)?

c. Use your answers from Part a and b to solve the inequality t 2−t−6≤0. Describe the solution using symbols and a number line graph.

d. Use similar reasoning to solve the inequalityt 2−t−6>0 and record the solutions using symbols and a number line graph.

2. Your answers to the questions in Problem 1 show how two key ideas reveal solutions to any quadratic inequality in the form a x2+bx+c≤0 or a x2+bx+c≥0.

a. How does sketching the graph off ( x )=a x2+bx+c help in solving a quadratic inequality in the form above?

b. How does solving the equation ofa x2+bx+c=0 help in solving a quadratic inequality like those shown?

3. What are the possible number of x-intercepts for the graph of a quadratic function? What are possible solution sets for quadratic inequalities? Sketch and briefly explain graphs to illustrate each possibility.

i. ii. iii.

To solve quadratic inequalities with one variable

1. Write the inequality with variables on the left and zero on the right.2. Use algebraic reasoning to locate the zeros.3. Graph the quadratic using the zeros and find the solution of the inequality.4. Record the solution using symbols.

4. Combine algebraic and graphic reasoning to solve the following inequalities. For each inequality:a. x2+2x>0

Solution:

b. x2+2x<0

Solution:

c. n2+2n−24≤0

Solution:

d. 3 x2−5 x>8

Solution:

e. −s2+2 s<6−2 s

Solution:

Practice1. Graph each quadratic inequality. Use algebraic reasoning to locate x-intercepts of the graph in order to solve the inequality. Record your solutions.a. k 2−3k>4

Solution:

b. −p2+10 p−7≤14

Solution:

c. −b2+8x ≥10

*you may use a calculator to get decimal value (nearest tenth) for x-intercept values.*

Solution:

2. The following graph shows how income and operating cost depend on ticket price and how they are related to each other. The relationship between income and operating cost

of a business are shown in the graph below.a. Use the graph to estimate answers for the following questions, and explain how you arrive at each estimate.i. Write the inequality that represents when the operating cost will exceed income?

ii. Write the inequality that represents when the income will exceed operating cost?

REVIEW3. Rewrite each expression with the smallest possible integer under the radical sign.a. √84 b. √160 c. 25√18

d. (3√5 )2 e. √ 1259 f. √20a2

4. Find the discriminant, then check the appropriate column for each equation.EQUATION DISCRIMINANT ONE

RATIONAL ROOT

TWO RATIONAL ROOTS

TWO IRRATIONAL ROOTS

NO REAL ROOTS

a. x2+6 x+10=0b. x2−4 x+1=0c. x2+6 x+9=0d. 4 x2−8=0e. −5 x2−10 x=0f. −x2=4 x+6g. 8 x2+2=8 xh. 7 x2+16 x+11=0

Hampton math · Web viewI. Simplifying Radicals, Solving Quadratics by Square Roots, and Complex...

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