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Accelerated CCGPS…Math II Name________________________ Intro to Complex Numbers Period_____Date_______________ Review: If we want to solve an equation, then we must work with both sides to “undo” what was done to the variable. For example, if x2 = 64, how do we “undo” squaring a number – we’ll take the square root on both sides. So, x2 64 , and the square root of 64 is 8. However, -8 is also a square root of 64, since (-8)(-8) = 64; hence, this equation actually has two solutions, which are oftentimes written as ±8. Example: (Sometimes we may need to use our rules for simplifying radicals as well.) Solve: 5 x 2 4 64 5 x 2 4 4 64 4 Subtracting 4 from both sides to get: 5x 2 60 5 5 Dividing both sides by 5: x2 12 5 x 2 60 x 2 12 x Taking the square root on each side: 4 3 2 3 Solve the following equations by “undoing”: 1.) x2 9 4.) x 7.) 2 36 0 2x 2 2 2.) x 2 144 3.) x 5.) x 2 1 0 6.) x 2 8 0 8.) 4 x 2 36 9.) 1 2 x 32 2 x 2 2 7 12.) 16 x 10.) x 2 3 1 11.) 13.) 3x 2 1 5 14.) 1 2 x 5 32 3 2 128 15.) 2 x 2 2 9 11 x 2 5 From our science class, we know that the time it takes an object to hit the ground when it is dropped from a height of s feet can be modeled by the equation h 16t 2 s , where t is the time in seconds. Find the time it takes when the object is dropped from the following heights: 16.) 80 feet 17.) 160 feet 18.) 320 feet New Idea: Not all equations will have real number solutions. For instance, x2 = -1 has no real number solutions because the square of any real number is never negative. To overcome this situation, we can expand our system of numbers to include the imaginary unit, defined as 1 i . (Note that i 2 = -1.) A complex number, written in standard form, is the number a + bi, wher a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part. Solve the following equations by “undoing”: 19.) x 2 16 20.) x 2 81 21.) x 2 144 0 22.) x 2 5 4 23.) x 2 1 3 24.) x 2 7 4x 2 5 Perform the indicated operations to write the expression as a complex number in standard form: 25.) 5 3i 2 4i 26.) 3 2i 1 i 27.) 7 2i 3 3i 28.) 5 i 3 8i 29.) i 11 5i 31.) i 4 i 30.) i 6 i 4 2i 32.) 3i 1 2i 33.) 4i 3 7i 34.) 1 3i 1 i 35.) 5 i 1 2i 36.) 2 3i 3 4i 37.) 3 2i 5 8i 38.) 2 4i 3 6i 39.) 1 1 2 i 2i 3 2 3 40.) 4 2i 1 5i 41.) 5 8i 2 9i 42.) 1 2 2 1 i i 2 3 3 4 43.) 5 4i 3 6i 44.) 2 5i 45.) 4 8i 4 8i 46.) 1 i 2 2