Functions and graphsAdditional / Further Maths

Solving equations that are quadratic in - by substitution

Use to solve a quadratic, then substitute back to find real values of .

  • Let , so .
  • Solve the resulting quadratic equation in , then replace with .
  • Take both square roots of a positive value. For , discard a negative value of .

Example 1

Substitute, solve and replace

Solve by substituting : , for .

Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .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

Check the real-number condition

Solve by substituting : , for .

Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .Discard this branch for real solutions: a real square cannot be negative.

Your turn

Question 1

Solve by substituting , for .

Check answer
Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .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 by substituting , for .

Check answer
Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .Discard this branch for real solutions: a real square cannot be negative.
Question 3

Solve by substituting , for .

Check answer
Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .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 by substituting , for .

Check answer
Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .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 by substituting , for .

Check answer
Introduce so the fourth-degree equation becomes quadratic in .Let .Factorise the expression.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.Replace the temporary variable by its definition, retaining each branch.Replace by .Discard this branch for real solutions: a real square cannot be negative.
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