Equation in quadratic form

Definition

An equation is in quadratic form if it can be written as $a(x^2 + bx + c) = 0$ where $a, b,$ and $c$ are constants. It often involves substituting a variable to simplify into a standard quadratic equation.

5 Must Know Facts For Your Next Test

1. Quadratic form equations can be solved by substitution to transform them into standard quadratics.
2. The general format is $A(u)^2 + B(u) + C = 0$ where $u$ is an expression involving the original variable.
3. Common methods for solving include factoring, completing the square, and using the quadratic formula.
4. Recognizing the structure of a quadratic form is crucial for correctly applying solution methods.
5. Solutions may sometimes involve finding the roots of the substituted variable first before back-substituting.

Review Questions

• What substitution would you use to solve $x^4 - 5x^2 + 6 = 0$?
• How do you recognize an equation that can be put into quadratic form?
• What methods can be used to solve equations once they are in quadratic form?

Related terms

Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$ where $a \neq 0$. Solutions can be found using various algebraic methods.

Substitution Method: A technique used to simplify an equation by replacing variables with another expression or variable.

Completing the Square: A method used to solve quadratics by rewriting them in perfect square trinomial form.