Factoring quadratics a first pass for a test

What is a quadratic equation?

An equation of the form $\displaystyle ax^2 + bx + c = 0$, where $\displaystyle a \neq 0$.

Example of a quadratic equation.

$\displaystyle 2x^2 + 4x - 6 = 0$ is a quadratic equation.

Factoring method for $\displaystyle x^2 - 5x + 6$?

Look for two numbers that multiply to 6 and add to -5. The factors are $\displaystyle (x - 2)(x - 3)$.

True or false: All quadratics can be factored.

False, because some quadratics have irrational or complex roots.

Fill in the blank: $\displaystyle x^2 + 7x + 10$ factors to ______.

$\displaystyle (x + 2)(x + 5)$

Difference between factoring and expanding.

Factoring breaks down expressions; expanding multiplies them out.

Question: How to factor $\displaystyle x^2 - 9$?

It can be factored as $\displaystyle (x - 3)(x + 3)$, a difference of squares.

What does the 'a' represent in $\displaystyle ax^2 + bx + c$?

'a' is the coefficient of $\displaystyle x^2$, determining the parabola's direction.

Scenario: Rent is $\displaystyle x$ for 3 months. What is total cost?

Total cost = $\displaystyle 3x$. If $\displaystyle x = 600$, total is $1800.

True or false: The quadratic formula always provides rational roots.

False, because roots can be irrational or complex.

What are roots of $\displaystyle x^2 + 4x + 4$?

The roots are $\displaystyle -2$ with a double root, from $\displaystyle (x + 2)^2 = 0$.

Fill in the blank: $\displaystyle x^2 - 5x + 6$ factors to ______.

$\displaystyle (x - 2)(x - 3)$

Scenario: You have $\displaystyle x$ apples, and $\displaystyle x - 2$ are bad. How many good?

Good apples = $\displaystyle x - (x - 2) = 2$.

Quiz question: What is the sum of roots for $\displaystyle x^2 + 6x + 8$?

Sum = $\displaystyle -b/a = -6/1 = -6$.

Comparison: Monomial vs. binomial.

A monomial has one term; a binomial has two terms.

How to factor $\displaystyle 2x^2 + 8x$?

Factor out the greatest common factor: $\displaystyle 2x(x + 4)$.