      UNIT 3  : QUADRATIC FUNCTIONS & EQUATIONS

LESSON 6:  COMPLEX NUMBERS DEFINED AS NON REAL ROOTS OF QUADRATIC EQUATIONS Solving Quadratic Equations with Non-Real Roots:

Example 1:   Solve the quadratic equation: Previously we would say this equation has no real roots because of the negative sign under the square root sign.

Definition: We could now write the roots as follows: These creatures with the i factor are called imaginary numbers.  We use them when solving quadratic equations with non real roots.

Example 2:   Solve 4x2 – 2x + 3 = 0

Solution: The roots of this equation are examples of Complex Numbers.

Note that complex roots always occur in pairs called complex conjugates.

The Complex Number system is the union of the set of all Real Numbers with the set of all Imaginary numbers (contain “i” factor)

It is defined formally as follows:  Example 3:    Solve 3x2 – 4x + 10 = 0  where x is a complex number.  Round roots to nearest hundredth.

Solution:  Example 3:  Find the quadratic equation in factored form whose roots are –4, 3

Solution:

(x - (-4))(x – 3) = 0

(x + 4)(x – 3) = 0

Example 4:  Find the quadratic equation in factored form if one root is 2 + 3i

Solution:

Since complex numbers occur in conjugate pairs, the other root is 2 – 3i.  Hence the equation is

[x - (2 + 3i)][x – (2 – 3i)] = 0    or

(x – 2 – 3i)(x – 2 + 3i) = 0