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Name Period ALGEBRA 2 Unit 2 – Solving Quadratics Completing the Square Essential Question: “ “ Perfect Square Trinomials Factor each expression. 𝑥 2 + 10𝑥 + 25 = 𝑥 2 + 18𝑥 + 81 = Observation 𝑏 2 𝑎𝑥 + 2(√𝑐)𝑥 + 𝑐 = 𝑎𝑥 + 𝑏𝑥 + ( ) 2 2 2 Middle coefficient = End term = What number would complete the square giving us a perfect trinomial? 𝑥 2 + 14𝑥 + _____ = 𝑥 2 + 24𝑥 + __________ = 3) 𝑥 2 + 3𝑥 + ___________ = 0 Completing the Square Given the standard form vertex form of the equation , we complete the square to find the Completing the Square 1) Move _________________________ to the other side of the equation 2) ______________________ to both sides of the equation 3) Find ____________ and place in each blank 4) ________________ left side of the equation and _______________ right side of the equation Factors 5) ________________ factors. (one factor squared) 6) Take the ______________________ of both sides 7) ______________ for variable 1. Solve by completing the square. 𝑥 2 − 2𝑥 − 5 = 0 Move c to the opposite side and add blanks. Fill in the blanks by completing the square. Factor the trinomial. Condense the factors. Solve for x. 2. Solve by completing the square. 𝑥 2 − 12𝑥 = −5 Completing the Square when 𝑎 ≠ 1 When there is a leading coefficient, still move c to the opposite side, but then we must factor the coefficient out. This adds an extra step! Solve by completing the square. 4𝑥 2 = 3 − 4𝑥 Complex Numbers Discovery Use your book (pg. 274-275) to define the following words and to answer any of the questions. Imaginary Number - i= 𝑖2 = How can I rewrite the square root of a negative number? Example Rewrite √−4 using imaginary numbers. Complex Number - What is the standard form for a complex number? Absolute Value of a Complex Number - |𝑎 + 𝑏𝑖| = Complex Numbers Essential Question: “ “ Imaginary Numbers 𝒊= 𝒊𝟐 = 𝒊𝟑 = 𝒊𝟒 = * Imaginary numbers are * When simplifying imaginary numbers: 𝑖5 = 𝑖7 = 𝑖 12 = To convert imaginary numbers: Divide exponent by four Remainder will tell us the power of i o Remainder of 1 = o Remainder of 2 = o Remainder of 3 = o Remainder of 4 = Examples: 1. 𝑖 63 = 2. 𝑖 120 = 3. 𝑖 7 = 4. 𝑖 36 = 5. 𝑖 15 = 6. 𝑖 86 = Simplify the following expressions. 1. 𝑖 4 + 𝑖 8 + 1 = 2. 2𝑖 − 6𝑖 2 + 5𝑖 14 + 7𝑖 9 = 3. 10𝑖 2 + 4𝑖 4 + 6 = 4. 12𝑖 − 10𝑖 22 + 8𝑖 16 − 6𝑖 33 + 16𝑖 80 = Complex Numbers Standard Form: Examples of Complex Numbers: Square Root of a Negative Number For any number, n, √−𝒏 = 𝒊√𝒏 Rewrite the following negative roots as roots with imaginary numbers. √−4 = √−13 = √−25 = Complex Number Operations Essential Question: “ “ Solve each of the equations. 1) (5𝑥 + 4)(3𝑥 + 12) = 2) (7 − 13𝑥) + (9 + 16𝑥) = 3) (4 + 7𝑥) − (12 − 16𝑥) = 4) (4𝑥)(7𝑥) = 5) (6𝑥 + 4)(3𝑥 + 10) = Adding Complex Numbers 1) (5 + 7𝑖) + (−2 + 6𝑖) = 2) (10 − 2𝑖 2 ) + (−8 + 4𝑖 2 ) = 3) (5 + √−13) + (8 + √−12) = Subtracting Complex Numbers 1) 7 – (3 + 2i) 2) (4 – 6i) – (2 + 3i) 3) (12 + 6i) – 10i Multiplying Complex Numbers 1) (5i)(-4i) 2) -6(2 – 4i) 3) 2i(3 – 6i) 4) (2 + 3i)(3 + 5i) 5) (5 – 2i)(-4 + i) Finding Complex Solutions Get the Take the Simplify the 1) 4𝑥 2 + 100 = 0 by itself of each side 2) 𝑥 2 = −16 Quadratic Formula Essential Question: “ “ The solution for a quadratic equation is values of x that SOLUTION = . = Quadratic Formula The equation must be in the standard form Standard form: X= a– b– c– Solve 3𝑥 2 + 5𝑥 + 1 = 0 using the quadratic formula. a= b= c= Substitute a, b, and c into the quadratic formula. 𝑥= −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 = Solve 𝑥 2 − 𝑥 − 1 = 0 using the quadratic formula. Solve 3𝑥 2 + 2𝑥 + 4 = 0 using the quadratic formula. Discriminant Discriminant = Value of the Discriminant Type and number of solutions for 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 𝑏 2 − 4𝑎𝑐 > 0 𝑏 2 − 4𝑎𝑐 = 0 𝑏 2 − 4𝑎𝑐 < 0 Determine the type and number of solutions for 𝑥 2 − 6𝑥 + 7 = 0. a= b= c= Substitute a, b, and c into discriminant. Determine the type and number of solutions for 4𝑥 2 + 2𝑥 + 3 = 0.