basic_engineering_mathematics0

10
Quadratic equations
10.1 Introduction to quadratic
equations
As stated in Chapter 7, 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 = 0isa
quadratic equation.
There are four methods of solving quadratic equations.
These are: (i) by factorization (where possible)
(ii) by ‘completing the square’
(iii) by using the ‘quadratic formula’
or (iv) graphically (see Chapter 13).
10.2 Solution of quadratic equations by
factorization
Multiplying out (2x + 1)(x − 3) gives 2x 2 − 6x + x − 3, i.e.
2x 2 − 5x − 3. The reverse process of moving from 2x 2 − 5x − 3
to (2x + 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 2x 2 − 5x − 3 = 0, then, by factorizing:
(2x + 1)(x − 3) = 0
Hence either (2x + 1) = 0 i.e. x =− 1 2
or (x − 3) = 0 i.e. x = 3
The technique of factorizing is often one of ‘trial and error’.
Problem 1. Solve the equations (a) x 2 + 2x − 8 = 0
(b) 3x 2 − 11x − 4 = 0 by factorization.
(a) x 2 + 2x − 8 = 0. The factors of x 2 are x and x. These are
placed in brackets thus: (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 added to the
product of the two outer terms must equal the middle term,
+2x in this case.)
The quadratic equation x 2 + 2x − 8 = 0 thus becomes
(x + 4)(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 + 4) = 0 i.e. x =−4
or (x − 2) = 0 i.e. x = 2
Hence the roots of x 2 + 2x − 8 = 0arex =−4 and 2
(b) 3x 2 − 11x − 4 = 0
The factors of 3x 2 are 3x and x. These are placed in brackets
thus: (3x )(x )
The factors of −4 are −4 and +1, or +4 and −1, or −2
and 2.
Remembering that the product of the two inner terms added
to the product of the two outer terms must equal −11x, the
only combination to give this is +1 and −4, i.e.
3x 2 − 11x − 4 = (3x + 1)(x − 4)