Evidence of Understanding

**algebraically determine the rational roots of a factorable polynomial (primarily quadratics)**- notice patterns that make a quadratic equation factorable or un-factorable
- describe the relationship between a quadratic equation’s roots and factors
- show a quadratic is factorable when there is no constant remaining and the original form of the equation is evenly divisible by either factor

- identify perfect squares and difference of squares (including even degree polynomials equations)
- recognize when factors correspond to imaginary roots

- recognize when factors correspond to imaginary roots

- notice patterns that make a quadratic equation factorable or un-factorable
**describe irrational, imaginary, or complex roots of a quadratic equation**- explain when a root is imaginary or irrational using a function’s degree and its graph or table
- approximate the value of an irrational root by analyzing a function’s graph or table
- calculate irrational,
**imaginary**, or**complex**roots from an equation in standard or vertex form- simplify a radical and use perfect squares to predict its location on a number line
- simplify a radical that involves an imaginary factor

- verify both roots must be imaginary if one root is imaginary (also verify for complex)

**generalize characteristics about the roots of any quadratic equation and use them to solve problems**- explain why perfect square trinomials have one unique root
- justify when
**completing the square**is an ideal strategy for identifying roots- create visual models for a given quadratic equation
- connect parts of a visual model with terms or quantities in its quadratic equation
- derive the
**quadratic formula**from the process of completing the square

- describe roots of a quadratic equation using the
**discriminant** - use given roots to describe features of the parabola or create a viable quadratic equation
- add, subtract, or multiply imaginary or complex numbers and explain the process

- determine all the roots using a variety of methods including analysis of a graph or table in addition to completing the square, factoring, or the quadratic formula

Develop conceptual understanding:

imaginary, complex, completing the square, quadratic formula, discriminant

Supporting terms to communicate:

roots, real, rational, irrational, perfect square, factor, vertex form, standard form, factored form, inverse operations