UP:MA19.A1.9

Students know:

Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
The method for completing the square.
Notational methods for expressing complex numbers.
A quadratic equation in standard form (ax²+bx+c=0) has real roots when b²-4ac is greater than or equal to zero and complex roots when b²-4ac is less than zero.

Students are able to:

Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Students understand that:

Solutions to a quadratic equation must make the original equation true and this should be verified.
When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
Different procedures for solving quadratic equations are necessary under different conditions.
If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax²+bx+c=0).