Unit 6 (Part II) – Triangle Similarity
Cholkar MCHS MATH II ___/___/___ Name____________________________
|U5L2INV2 |How is the technique of completing the square used to derive the quadratic formula? |
| |How does the quadratic formula suggest the need for new kinds of numbers (imaginary numbers)? |
|HW # | CYU pg. 356; pg. 358 # 4; pg. 363 # 27 [CYU, 4, 27] |
|Do Now |Simplify the following radicals. |
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| |a. [pic] b. [pic] c. [pic] d. [pic] |
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INVESTIGATION: THE QUADRATIC FORMULA AND COMPLEX NUMBERS
(Adapted from Core-Plus 3 pg. 353)
My role for this investigation _________________________
As you have seen in your previous studies, when problems require solving quadratic equations that cannot be factored easily, you can always turn to the quadratic formula. For instance, solutions for
are given by
As you complete the problems in this investigation, look for answers to these questions:
PROVING AND USING THE QUADRATIC FORMULA
The work that you have done to write quadratic expressions in vertex form is closely related to the quadratic formula that can be used to find solutions of any quadratic equation.
1. Consider the general form of a quadratic equation,
The solutions of this equation are given by and
Explain how each step in the following derivation of the quadratic formula is justified by properties of numbers and operations.
[pic] ________________________________________________________
Then [pic] ________________________________________________________
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2. The quadratic formula provides a tool for solving any quadratic equation by algebraic reasoning. But you have other helpful strategies available through use of technology.
a. How could you use calculator- or computer-generated tables of function values or graphs to estimate solutions for a quadratic equation like [pic]
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3. Use the quadratic formula to solve each of the following equations. Report your answers in exact form, using radicals where necessary rather than decimal approximations. Check each answer by substituting the solution values for x back into the original equation.
a. [pic] b. [pic] c. [pic]
d. [pic] e. [pic] f. [pic]
4. Solutions for problem 3 involved several kinds of numbers. Some could be expressed as integers. But others could only be expressed as fractions or as irrational numbers involving radicals. One of the quadratic equations appears to have no solutions.
a. At what point in use of the quadratic formula do you learn whether the equation has two distinct solutions, only one solution, or no real solutions?
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b. If the coefficients of a quadratic equation are integers or rational numbers, at what point in use of the quadratic formula do you learn whether the solution(s) will be integers, rational numbers, or irrational numbers?
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COMPLEX NUMBERS
When trying to solve the quadratic formula gives
5. Sketch a graph of the function Explain how it shows that there are no real number solutions for the equation
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When discriminate is negative, then the quadratic equation
has no real number solutions.
The obstacle to solving was removed by reasoning like this.
Complex Numbers: In the form [pic]
Imaginary Number: [pic] is replaced by the letter [pic]
6. 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 where a and b are real numbers.
a. [pic] b. [pic] c. [pic]
d. [pic]
|Lesson Summary |In this investigation, you used the technique of completing the square to derive the quadratic formula and practiced use of |
| |that formula in solving quadratic equations. |
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| |[pic] |
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| |[pic] |
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| |[pic] |
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Math Toolkit Vocabulary: quadratic formula, complex numbers, imaginary numbers, discriminate[pic]
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