# Quadratic Equations

**Definition 1.1. **A quadratic equation is an equation in the form

ax^{2} + bx + c = 0,

where a, b, and c are real numbers, and a is nonzero. “a” is the coefficient
of

the quadratic term (or, is the leading coefficient), “b” is the coefficient of

the linear term, and c is the constant term.

**Remark 1.2.** Officially, the only tool we have to use in solving
quadratic equations

is factoring. There are some quadratics, however, that do not factor nicely,

or the factorization does not come to mind easily.

**Definition 1.3.** The method of solving a quadratic equation know as
completing

the square involves transforming a quadratic into a perfect square.

**Remark 1.4.** Add picture here for geometric interpretation.

**Item 1.5.** We look for a pattern in the square of binomials:

Notice that the linear term is twice the constant in the binomial we are
squaring.

Moreover, notice that the constant term of the quadratic is the square of

the binomial’s constant. Therefore, when we complete the square, we want the

constant term to be the square of one-half the linear term.

**Example 1.6.** Solve the following quadratic by completing the square:

x^{2} − 4x + 3 = 0.

**Soln:** In order to solve by completing the square, we wish to move the

constant term out of the way:

x^{2} − 4x = −3.

Now, in our list above, we noticed that perfect squares have the constant
term

as the square of half the middle term. Thus, we add it to both sides of the

equation:

Again, from our list above, we see that the square quadratic factors as the
square

of a sum of x and half the linear term:

(x − 2)^2 = 1

Solving for x, we take the square root of each side:

**Example 1.7. **Solve the following quadratic by completing the square:

x^{2} − x − 1 = 0.

**Soln: **Following the same steps as above we have:

where
is the **Golden Ratio.**

**Remark 1.8.** Completing the square will only work if the leading
coefficient,

that is, the coefficient of the quadratic term, is one. What can we do if it is

not?

**The Quadratic Formula** Consider the general quadratic equation

ax^{2} + bx + c = 0

We wish to complete the square, so we must have a leading coefficient of one.

Divide everything by “a”:

Clear the constant term as before:

Again, we wish to have our new constant term the square of half the linear term:

Simplifying the left side, and finding a common denominator for the right, we

have

Taking the square root of each side gives

And solving for x:

**Definition 1.9. **The quadratic formula is an algorithm by which to find
the

roots of a quadratic equation. The two roots of an equation of the form

ax^{2} + bx + c = 0

are

**Example 1.10**. Find the roots of the quadratic equation:

x^{2} = 5 − 3x

**Soln:** First, we must put the equation in the correct form:

x^{2} + 3x − 5 = 0.

Using the quadratic formula, with a = 1, b = 3, c = −5, we have:

**Definition 1.11.** The discriminant is the term underneath the radical
in the

quadratic formula, namely, b^2 − 4ac.

**Remark 1.12.** If the discriminant is positive, we will have two real
roots. If the

discriminant is zero, we will have one real root. If the discriminant is
negative,

we have no real roots.

**Item 1.13.** Please to eight of the following problems in section 3.1
for Friday:

1, 3, 5, 7, 13, 14, 27, 29, 37, 38, 47, 48, 55, 63, 65, 77

## 2 The Graph of a Quadratic Function

**Remark 2.1.** Since a quadratic is characterized by having an x^{2} term,
quadratics

can be shown to be reflections, translations, and stretches of the base function

y = x^{2}.

**Example 2.2.** Graph the quadratic function y = x^{2} − 4x + 3.

**Soln:** By completing the square, we will be able to shift the graph of
x^{2} to

the graph of y :

So our function is x^{2} shifted two units to the right and one unit down.
Notice

that the lowest point of x^{2}, (0, 0) is moved to (2,−1).

**Definition 2.3**. The turning point that occurs in a parabola, the graph
of a

quadratic function, is called the vertex of the function. Since the base
quadratic

is an even function, all quadratics have an axis of symmetry, the vertical line

passing through the vertex. If the parabola opens upwards, the vertex will be

the minimum of the function, and if it opens upwards, the vertex will be the

maximum.

**Definition 2.4.** A dilation or stretching of a quadratic will affect
how widely

the parabola opens.

**Example 2.5**. Graph the function y = 3x^{2} + 6x + 3.

**Soln: **Again, we’ll complete the square to see how the graph of x^{2}
moves:

So x^{2} is shifted one to the left and then “multiplied by three.” Notice that

points like (0, 3) and (2, 27) are on the graph of 3(x + 1)^{2} instead of the
points

(0, 1) and (2, 9) on the graph of (x_{1})^{2}. Thus, the multiplying by three makes the

graph steeper, or skinner.

**Remark 2.6.** If a quadratic function is in the form

then the vertex of the graph is at (h, k) and the value of
a characterizes how

“fat/skinny” the graph is: if a > 0, the graph opens upward. If a < 0, the graph

opens downwards. If |a| > 1, the graph gets skinner, and if |a| < 1, the graph

spreads out.

**Example 2.7. **Graph the function y = −2x^{2} + 12x −
16.

**Soln:** Again, we complete the square. Remember that
the leading coefficient

must be one for this to work:

The “+18” balances out the “−2 ยท +9” inside the parentheses.

So the graph of y looks like the graph of x^{2} moved three
units to the right,

flipped over the x-axis, made skinner, and shifted vertically two units up.

**Remark 2.8.** Remember that the maximum or minimum of
a quadratic always

occurs at the vertex. Since we are using complete the square, the vertex can
also

be written as

**Example 2.9.** We saw above that the vertex of y =
−2x^{2} + 12x − 16 is (3, 2).

We can get that directly by seeing that the vertex is at (−b/2a, f(−b/2a))

So the vertex of y is at (3, 2).

**Item 2.10.** The following homework problems are not
assigned, but it is

highly recommended that you do them, for help on the test. For section

5.2, please do

1, 5, 7, 13, 19, 21, 23, 25, 27, 31, 33 − 38, 39, 41, 51.