Chapter 14
Solving quadratic equations
14.1 Introduction
As stated in Chapter 11, an equation is a statement
that two quantities are equal and to ‘solve an equation’
means ‘to find the value of the unknown’. The value of
the unknown is called the root of the equation.
A quadratic equation is one in which the highest
power of the unknown quantity is 2. For example,
x 2 − 3x + 1 = 0 is a quadratic equation.
There are four methods of solving quadratic equations. These are:
(a) by factorization (where possible),
(b) by ‘completing the square’,
(c) by using the ‘quadratic formula’, or
(d) graphically (see Chapter 19).
14.2 Solution of quadratic equations
by factorization
Multiplying out (x + 1)(x − 3) gives x 2 − 3x + x − 3
i.e. x 2 − 2x − 3. The reverse process of moving from
x 2 − 2x − 3 to (x + 1)(x − 3) is called factorizing.
If the quadratic expression can be factorized this
provides the simplest method of solving a quadratic
equation.
For example, if x 2 − 2x − 3 = 0, then, by factorizing
(x + 1)(x − 3) = 0
Hence, either
(x + 1) = 0, i.e. x = −1
or
(x − 3) = 0, i.e. x = 3
Hence, x = −1 and x = 3 are the roots of the quadratic equation x 2 − 2x − 3 = 0.
The technique of factorizing is often one of trial and
error.
Problem 1. Solve the equation x 2 + x − 6 = 0 by
factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −6 are +6 and −1, or −6 and +1, or +3
and −2, or −3 and +2.
The only combination to give a middle term of +x is
+3 and −2,
i.e.
x
2
+ x − 6 = (x + 3)(x − 2)
The quadratic equation x 2 + x − 6 = 0 thus becomes
(x + 3)(x − 2) = 0
Since the only way that this can be true is for either the
first or the second or both factors to be zero, then
either
(x + 3) = 0, i.e. x = −3
or
(x − 2) = 0, i.e. x = 2
Hence, the roots of x 2 + x − 6 = 0 are x = −3 and
x = 2.
Problem 2. Solve the equation x
2
+ 2x − 8 = 0
by factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −8 are +8 and −1, or −8 and +1, or +4
and −2, or −4 and +2.
The only combination to give a middle term of +2x is
+4 and −2,
i.e.
x
2
+ 2x − 8 = (x + 4)(x − 2)
(Note that the product of the two inner terms (4x) added
to the product of the two outer terms (−2x) must equal
the middle term, +2x in this case.)
DOI: 10.1016/B978-1-85617-697-2.00014-4
Solving quadratic equations
14.1 Introduction
As stated in Chapter 11, an equation is a statement
that two quantities are equal and to ‘solve an equation’
means ‘to find the value of the unknown’. The value of
the unknown is called the root of the equation.
A quadratic equation is one in which the highest
power of the unknown quantity is 2. For example,
x 2 − 3x + 1 = 0 is a quadratic equation.
There are four methods of solving quadratic equations. These are:
(a) by factorization (where possible),
(b) by ‘completing the square’,
(c) by using the ‘quadratic formula’, or
(d) graphically (see Chapter 19).
14.2 Solution of quadratic equations
by factorization
Multiplying out (x + 1)(x − 3) gives x 2 − 3x + x − 3
i.e. x 2 − 2x − 3. The reverse process of moving from
x 2 − 2x − 3 to (x + 1)(x − 3) is called factorizing.
If the quadratic expression can be factorized this
provides the simplest method of solving a quadratic
equation.
For example, if x 2 − 2x − 3 = 0, then, by factorizing
(x + 1)(x − 3) = 0
Hence, either
(x + 1) = 0, i.e. x = −1
or
(x − 3) = 0, i.e. x = 3
Hence, x = −1 and x = 3 are the roots of the quadratic equation x 2 − 2x − 3 = 0.
The technique of factorizing is often one of trial and
error.
Problem 1. Solve the equation x 2 + x − 6 = 0 by
factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −6 are +6 and −1, or −6 and +1, or +3
and −2, or −3 and +2.
The only combination to give a middle term of +x is
+3 and −2,
i.e.
x
2
+ x − 6 = (x + 3)(x − 2)
The quadratic equation x 2 + x − 6 = 0 thus becomes
(x + 3)(x − 2) = 0
Since the only way that this can be true is for either the
first or the second or both factors to be zero, then
either
(x + 3) = 0, i.e. x = −3
or
(x − 2) = 0, i.e. x = 2
Hence, the roots of x 2 + x − 6 = 0 are x = −3 and
x = 2.
Problem 2. Solve the equation x
2
+ 2x − 8 = 0
by factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −8 are +8 and −1, or −8 and +1, or +4
and −2, or −4 and +2.
The only combination to give a middle term of +2x is
+4 and −2,
i.e.
x
2
+ 2x − 8 = (x + 4)(x − 2)
(Note that the product of the two inner terms (4x) added
to the product of the two outer terms (−2x) must equal
the middle term, +2x in this case.)
DOI: 10.1016/B978-1-85617-697-2.00014-4