c

–

a

b

–

a

quadratic 429

c

–

e

p

–

d

q

–

e

a

–

d

q/e

—–

c/e

a/d

—–

p/d

q/e

—–

c/e

a/d

—–

p/d

c

–

e

q

–

e

p

–

d

a

–

d

1

–

2

3

–

4

b2

–

4a

b

–

2a

b

–

2a

The quantity b2–4ac is called the

DISCRIMINANT

of

the quadratic equation ax2+ bx + c= 0. The quadratic

formula shows that if the discriminant is a positive real

number, then the equation has two distinct real solutions.

If it is zero, then the equation has just one (double) root,

and if the discriminant is a negative real number, then

there are no real solutions to the equation. (There are,

however, two distinct complex solutions.) Given two

numbers r1and r2, then x2– (r1+ r2)x+ r1r2= 0 is a

quadratic equation with those two numbers as solutions.

In general, the quadratic formula shows that the two

roots of a quadratic equation ax2+ bx + c= 0 have sum

–and product .

A quadratic function is a function of the form f(x) =

ax2+ bx + cwith a≠0. If the constants a, b, and care

real numbers, then the graph of a quadratic function is

a

PARABOLA

,

CONCAVE UP

if ais positive and concave

down if ais negative. Rewriting the equation as:

shows that, if ais positive, the quadratic function has

minimal value when x= –(If ais negative, then this

produces a maximal value for the function.) The point

(–,c– ) represents the vertex of the parabola. The

graph crosses the x-axis twice if the quadratic equation

f(x) = ax2+ bx + c= 0 has two real solutions, touches

the x-axis at a turning point if the two solutions are

equal, and does not cross the x-axis at all if there are

no real solutions.

Students in high school are often taught to solve

quadratic equations by a process of factoring. One

notes that the product of two linear terms Ax + Band

Cx + Dyields a quadratic

(Ax + B)(Cx + D) = ACx2+ (AD + BC)x+ BD

with coefficients a= AC, b = AD + BC, and c= BD.

Here bis a sum of two factors of the product ac. This

suggests the following procedure:

To solve a quadratic equation ax2+ bx + c= 0,

select two factors pand qof the product ac that

sum to b. (This gives pq = ac and p+ q= b.)

Rewrite the quadratic equation as

ax2+ px + qx + c= 0

and perform algebra to factor the equation.

Begin by selecting a common factor of the first

two terms, ax2and px, and a common factor

of the final two terms, qx and c.

For example, to solve 8x2+ 2x– 3 = 0, we seek two

factors of –24 that sum to two. This suggests the fac-

tors –4 and 6. Rewriting, we obtain:

8x2+ 2x– 3 = 8x2– 4x+ 6x– 3

= 4x(2x– 1) + 3(2x– 1)

= (4x+ 3)(2x– 1)

It is now clear that the quadratic equation 8x2+ 2x– 3

= (4x+ 3)(2x– 1) = 0 has solutions x= –and x= .

It is surprising that if a, b, and care whole num-

bers, this method will never produce fractional coeffi-

cients. One can explain why this is the case using

techniques from

NUMBER THEORY

:

Given a quadratic expression ax2+ bx + cwith

integral coefficients, and integers pand qwith

pq = ac and p+ q= b, let dbe the

GREATEST

COMMON DIVISOR

of aand p, and ethe greatest

common divisor of qand c. Then the numbers

and are integers sharing no common

factors. The same is true for and . Since

ac = pq we have = . The fraction is

presented in reduced form, as is the fraction

. Since they are the same reduced fraction,

their numerators and denominators match.

Consequently, = and = . Factoring

the quadratic expression now yields: