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 Dependent Variable

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# Quadratic Formula

## 1 Introduction

The main intention of the quadratic formula is to find solutions to equations of the form where a, b, and c are real numbers with a ≠ 0. The quadratic formula tells you that, in
general, you have two such solutions and those solutions are: Actually you can use the quadratic formula for just about anything involving polynomials
of degree 2 as long as you know how to look at it in the right way. We will use the quadratic
formula for about three different types of problems:

(1) Factoring polynomials of the form ax^2 + bx + c. [Sections 6.2-6.5]
(2) Solving equations of the form ax^2 + bx + c = 0 [Sections 6.6, 12.1, 12.2]
(3) Graphing equations of the form y = ax^2 + bx + c [Section 12.5]

## 2 Problem Type 1 - Factoring

Factoring, non-trivially, a quadratic means writing a polynomial of degree 2 as a product of
two polynomials of degree 1. For example taking The real problem is how do you find the numbers for p, q, r, and t which will actually
work. There are many trial-and-error methods which you can employ. With a little practice,
for simple examples these can be quite fast. This shows the merit in the methods demonstrated
in Sections 6.2-6.5 of our book. However, there is a way of computing these in a
straightforward manner.

We are actually able to factor any quadratic, but the numbers for p, q, r, and t might
not be "nice". In our case, "nice" means that they are integers (whole numbers). If they are
not, the we say that quadratic is prime or irreducible.

The direct way of finding the p, q, r, and t which will factor is actually given by the
Quadratic Formula and the process involves three steps (actually 4, but step 0 is only a prep
stage that you won't have to do all the time). The steps are:

(0) Take out any factors which are common to all terms.

(1) Compute the discriminant If the discriminant D is negative or not a perfect square, then the quadratic is prime
and you stop. You can easily check whether or not D is a perfect square using your
calculator.

(2) Find the roots from the quadratic formula: (3) Re-write the roots as factors. From the formulas, the roots will look like fractions
x = p/q. In order to re-write them as factors, the denominator becomes the coefficient
of x and you subtract the numerator. Important Idea. Remember that the important thing is getting the correct association of
a, b, and c. If the term is being subtracted, then remember the coefficient is negative. The
correspondence is: Example 1. Factor 20x^2 + 21x - 54.

Solution. Identifying a, b, and c, we have Computing the discriminant We can use the calculator as a check to see if 4761 is a perfect square. Using the -button
on the calculator, we have a whole number, so it will factor

Calculating the roots  Thus, our factorization is Checking this Example 2. Factor x^2 + 14x + 48.
Solution. We first have for a, b, and c Then calculating the discriminant We then have or Thus Example 3. Factor x^2 - 2x + 10.
Solution. Identifying a, b, and c, we have Calculating the discriminant negative, so won't factor

Thus, Example 4. Factor Solution. Identifying a, b, and c Calculating the discriminant Thus, Example 5. Factor Solution. Identifying a, b, and c Calculating the discriminant Calculating the roots and factors, Example 6. Factor x^2 - 81.
Solution. We actually have so identifying a, b, and c Calculating the discriminant (will factor):

Calculating the roots and factors, we have Actually, the difference of squares may be one of the few times the special formulas from
section 6.5 are really useful as a short-cut in factoring. We have So in the previous example, if we recognize that 81 is a perfect square 81 = 9^2 then we have immediately. But as always, this is a short-cut and not really necessary.

Example 7. Factor Solution. We first notice that each term has a factor of 2x in common. This means we have So we use the quadratic formula to factor 4x^2 + 4x - 3. Identifying a, b, and c Calculating the discriminant Calculating the roots and factors, Thus, for our final answer we have Example 8. Factor Solution. The idea is the same as before, but we treat y just as if it were a number. Identifying
a, b, and c Calculating the discriminant Note that Calculating the roots and factors, 