What are complex roots?
- Complex numbers provide solutions for quadratic equations which have no real roots
- Complex roots occur when solving a quadratic with a negative discriminant
- This leads to square rooting a negative number
How do we solve a quadratic equation with complex roots?
- We solve an equation with complex roots in the same way we solve any other quadratic equations
- If in the form we can rearrange to solve
- If in the form we can complete the square or use the quadratic formula
- We use the property along with a manipulation of surds
- When the coefficients of the quadratic equation are real, complex roots occur in complex conjugate pairs
- If is a root of a quadratic with real coefficients then is also a root
- When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
- To solve these use the quadratic formula
How do we find a quadratic equation given a complex root?
- We can find the equation of the form if you are given a complex root in the form
- We know that the complex conjugate is another root,
- This means that andare factors of the quadratic equation
- Writing this as will speed up expanding
- Expanding and simplifying gives us a quadratic equation where b and c are real numbers
- Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation.
- You should get 0 if correct! [Note: 0 is equivalent to 0 + 0i]