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Chapter 3 Quadratic Functions and Equations

Chapter 3 Quadratic Functions and Equations. Complex Numbers. 3.3. Perform arithmetic operations on complex numbers Solve quadratic equations having complex solutions. Properties of the Imaginary Unit i.

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Chapter 3 Quadratic Functions and Equations

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  1. Chapter 3 Quadratic Functions and Equations

  2. Complex Numbers 3.3 Perform arithmetic operations on complex numbers Solve quadratic equations having complex solutions

  3. Properties of the Imaginary Unit i Defining the number i allows us to say that the solutions to the equationx2+ 1 = 0arei and –i.

  4. Complex Numbers A complex number can be written in standard form as a + bi where a and b are real numbers. The real part is a and the imaginary part is b. Every real number a is also a complexnumber because it can be written as a + 0i.

  5. Imaginary Numbers A complex number a + bi with b ≠ 0 is animaginary number. A complex numbera + bi with a = 0 and b ≠ 0 is sometimes calleda pure imaginary number. Examples of pure imaginary numbers include 3i and –i.

  6. The Expression If a > 0, then

  7. Example: Simplifying expressions Simplify each expression. Solution

  8. Example: Performing complex arithmetic Write each expression in standard form. Support your results using a calculator. a) (3 + 4i) + (5  i) b) (7i)  (6  5i) c) (3 + 2i)2 d) Solution a) (3 + 4i) + (5  i) = 3 + 5 + 4i  i = 2 + 3i b) (7i)  (6  5i) = 6  7i + 5i = 6  2i

  9. Example: Performing complex arithmetic c) (3 + 2i)2 = (3 + 2i)(3 + 2i) = 9 – 6i – 6i + 4i2 = 9  12i + 4(1) = 5  12i d)

  10. Quadratic Equations with Complex Solutions We can use the quadratic formula to solve quadratic equations if the discriminant is negative. There are no real solutions, and the graph does not intersect the x-axis. The solutions can be expressed as imaginary numbers.

  11. Example: Solving a quadratic Solve the quadratic equation x2 + 3x + 5 = 0.Support your answer graphically. Solution a = 1, b = 3, c = 5

  12. Example: Solving a quadratic Solution continued The graph does not intersect the x-axis, so no real solutions, but two complex solutions that are imaginary.

  13. Example: Solving a quadratic Solve the quadratic equationSupport your answer graphically. Solution Rewrite the equation: a = 1/2, b = –5, c = 17

  14. Example: Solving a quadratic Solution continued The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.

  15. Example: Solving a quadratic Solve the quadratic equation –2x2 = 3. Support your answer graphically. Solution Apply the square root property.

  16. Example: Solving a quadratic Solution continued The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.

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