- Idea
- 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:
- By factoring--doesn't always work since polynomials can not always be factored;
- By completing the square--always works; and
- 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....
- Rewrite the equation in descending order of the variable with one side of the equation equal to 0.
- Rewrite the variable in the highest-degree term as the square of the variable in the middle term.
- 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.
- Solve the equation au2 + bu + c = 0 by
- factoring,
- completing the square, or
- using the quadratic formula.
- Replace u with the expression of the original variable (that is, substitute original variable back in).
- Solve the resulting equation in the original variable.
- Check for extraneous solutions.
|
