Math 101 Intermediate Algebra

Writing Equations in Quadratic Form
Chapter 9, Section 3


Spot an equation that is in quadratic form, as done in Chapter 6, section 2.

Make a substitution that leaves a quadratic polynomial in one variable and solve it.

Resubstitute the original variable back in and solve for the original variable.

Always check the solutions for extraneous solutions since it's real easy to come up with solutions that don't solve the original equation.

Solving Quadratic Equations

You now have three ways to solve quadratic equations:

  1. By factoring--doesn't always work since polynomials can not always be factored;

  2. By completing the square--always works; and

  3. By using the quadratic formula--always works.

Choose the way that is easiest for the given problem.

Steps to Solving Equations in Quadratic Form

First, notice that the equation is in quadratic form, then....

  1. Rewrite the equation in descending order of the variable with one side of the equation equal to 0.

  2. Rewrite the variable in the highest-degree term as the square of the variable in the middle term.

  3. Make a substitution that will result in an equation of the form au2 + bu + c = 0, with a not zero and u standing for some expression in terms of the original variable.

  4. Solve the equation au2 + bu + c = 0 by

    • factoring,
    • completing the square, or
    • using the quadratic formula.

  5. Replace u with the expression of the original variable (that is, substitute original variable back in).

  6. Solve the resulting equation in the original variable.

  7. Check for extraneous solutions.