Polynomial functionsUnit 1

Solving equations that are quadratic in - real and complex solutions

Find all real and non-real complex solutions of an equation quadratic in .

  • Work in , which includes real numbers and non-real complex numbers.
  • Factor as a quadratic in and set each factor equal to zero.
  • If with , then . Keep both complex roots.

Example 1

Two real and two non-real roots

Solve , for .

Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.

Example 2

A different complex pair

Solve , for .

Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.

Your turn

Question 1

Solve, for .

Check answer
Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.
Question 2

Solve, for .

Check answer
Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.
Question 3

Solve, for .

Check answer
Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.
Question 4

Solve, for .

Check answer
Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.
Question 5

Solve, for .

Check answer
Factor the expression.Set each factor equal to zero, then isolate . Retain every distinct solution.Now use the other value from the quadratic, retaining both cases.Use : .Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.
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