A QUADRATIC is a polynomial

whose highest exponent is 2.

A quadratic equation is a second-order

polynomial equation in a single variable x

with a ≠ 0. Because it is a second-order

polynomial equation, the fundamental

theorem of algebra guarantees that it

has two solutions. These solutions may

be both real, or both complex.

ax2+bx+c=0

The roots can be found by completing the square,

Solving for then gives

This equation is known as the quadratic formula.

The plus-minus sign states that you

have two numbers

and

What do we mean by a root of a quadratic?

A solution to the quadratic equation.

For example, the roots of this quadratic

x² + 2x − 8

are the solutions to

x² + 2x − 8 = 0.

To find the roots, we can factor that quadratic as

(x + 4)(x − 2).

Now, if x = −4, then the first factor will be 0. While if x = 2, the second factor will be 0. But if any factor is 0, then the entire product will be 0. Therefore, if x = −4 or 2, then

x² + 2x − 8 = 0.

−4 and 2

are the solutions to the quadratic

equation. They are the roots of that

quadratic.

A root of a quadratic

is also called a zero. Because, as we will

see, at each root, the value of the graph is 0.

How many roots has a quadratic?

Always two. Because a quadratic (with leading

coefficient 1, at least) can always be factored

as (x − a)(x − b), and a, b are the two roots.

Note that if a factor is (x + q), then the root is

−q.

For, (x + q) can take the form (x − a):

(x + q) = [x − (−q)].

−q is the root,

What do we mean by a double root?

The two roots are equal. The factors are

(x − a)(x − a), so that the two roots are a, a.

For example, this quadratic

x² − 10x + 25

can be factored as

(x − 5)(x − 5).

If x = 5, then each factor will be 0, and

therefore the quadratic will be 0. 5 is called a

double root.

When will a quadratic have a

double root?

When the quadratic is a

perfect square

trinomial.

Find the roots of each

quadratic by factoring.

a) x² − 3x + 2

(x − 1)(x − 2)

x = 1 or 2.

b) x² + 7x + 12

(x + 3)(x + 4)

x = −3 or −4.

c) x² + 3x − 10

(x + 5)(x − 2)

x = −5 or 2.

d) x² − x – 30

(x + 5)(x − 6)

x = −5 or 6.

e) 2x² + 7x + 3

(2x + 1)(x + 3)

x = − 1 or −3.

2

f) 3x² + x − 2

(3x − 2)(x + 1)

x = 2 or −1.

3

g) x² + 12x + 36

(x + 6)²

x = −6, −6.

A double root.

h) x² − 2x + 1

(x − 1)²

x = 1, 1.

A double root.

Notice that we use

the conjunction "or,"

because x takes on

only one value at a

time.

c = 0. Solve this

quadratic equation:

ax² + bx = 0

Since there is no constant term -- c = 0 --

x is a common factor:

x(ax + b) = 0.

This implies:

x= 0or

x = b

a .

Those are the two roots.

a) x² − 5x

x(x − 5)

x = 0 or 5.

b) x² + x

x(x + 1)

x = 0 or −1.

c) 3x² + 4x

x(3x + 4)

x = 0 or −4

3

d) 2x² − x

x(2x − 1)

x = 0 or ½

b = 0. Solve this

quadratic equation:

ax² − c = 0.

In the case where there is no middle

term, we can write:

ax²=c.

This implies:

x²=c

a

x=

However, if the form is the difference of two

squares --

x² − 16

-- then we can factor it as:

(x + 4)(x −4).

The roots are ±4.

In fact, if the quadratic is

x² − c,

then we could factor it as:

(x + )(x − ),

so that the roots are ± .

a) x² − 3

x² = 3

x = ± .

b) x² − 25

(x + 5)(x − 5)

x = ±5.

c) x² − 10

(x + )(x − )

x = ± .

Solve each equation

for x.

a) x² = 5x − 6

x² − 5x + 6 = 0

(x − 2)(x − 3) = 0

x = 2 or 3.

b) x² + 12 = 8x

x² − 8x + 12 = 0

(x − 2)(x − 6) = 0

x = 2 or 6.

c) 3x² + x = 10

3x² + x − 10 = 0

(3x − 5)(x + 2) = 0

x = 5/3 or − 2.

d) 2x² = x

2x² − x = 0

x(2x − 1) = 0

x = 0 or 1/2.

Solve this equation

We can put this equation in the

standard form by changing all the signs

on both sides. 0 will not change. We

have the standard form:

3 −52

x − 3x² = 0

Next, we can get rid of the fraction by

multiplying both sides by 2. Again, 0

will not change.

6x² + 5x − 6 = 0

(3x − 2)(2x + 3) = 0.

The roots are23

and −32

THE END.