Polynomial functionsUnit 1

Solving equations that are quadratic in

Factor into two quadratic factors, then find all real solutions.

  • An equation is quadratic in .
  • Factor into two expressions involving , and set each factor equal to zero.
  • For , a positive value of gives two square roots. A negative value gives no real solution.

Example 1

Two quadratic factors

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.Take both square roots. Include positive and negative roots.Solve the other square-root branch as well, then collect all roots.Evaluate the plus and minus branches and list all solutions.

Example 2

Real solutions only

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.Discard this branch for real solutions: a real square cannot be negative.

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.Take both square roots. Include positive and negative roots.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.Take both square roots. Include positive and negative roots.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.Take both square roots. Include positive and negative roots.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.Take both square roots. Include positive and negative roots.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.Discard this branch for real solutions: a real square cannot be negative.
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