⇒ The quadratic equation ax² + bx + c = 0 has solutions given by:

⇒ If the expression under the square root is negative there are no real solutions

⇒ You can find solutions to the equation in all cases by extending the number system to include √-1

⇒ Since there is no real number that squares to produce -1, the number √-1 is called an imaginary number and is represented using the letter i

⇒ Sums of real and imaginary numbers, for example 3 + 2i are known as complex numbers

⇒ In a complex number, the real part and the imaginary part cannot be combined to form a single term

• Complex numbers can be added or subtracted by adding or subtracting their real parts and adding or subtracting their imaginary parts
• You can multiply a real number by a complex number by multiplying out the brackets in the usual way

⇒ Note: complex numbers are often represented by the letter z or w

⇒ You can use complex numbers to find solutions to any quadratic equation with real coefficients

• If b² - 4ac < 0 then the quadratic equation ax² + bx + c = 0 has two distinct complex roots, neither of which are real

⇒ Also see our notes on:

• Multiplying Complex Numbers
• Complex Conjugation
• Roots of Quadratic Equations
• Solving Cubic and Quartic Equations