Algebra26P1^
1
Take the coefficient of x: (^) +ba
Halve it: +b
2 a
Square the answer: +b
a
2
42
Add it to both sides of the equation:
⇒ (^) x bx
a
b
a
b
a
c
a
2 2
2
2
++ 44 = 2 –
Factorise the left-hand side and tidy up the right-hand side:
⇒ x ba bac
a
()+ =
2
4
4
(^22)
2
–
Take the square root of both sides:⇒ x b
abac
a+=±
2
4
2
(^2) –
(^)
⇒ x=––bb± a ac
(^24)
2
This important result, known as the quadratic formula, has significance beyond
the solution of awkward quadratic equations, as you will see later. The next two
examples, however, demonstrate its use as a tool for solving equations.
EXAMPLE 1.32 Use the quadratic formula to solve 3x^2 − 6 x + 2 = 0.
SOLUTION
Comparing this to the form ax^2 + bx + c = 0
gives a = 3, b = –6 and c = 2.
Substituting these values in the formula x bb ac
a
=––±
(^24)
2
gives x=^63 ± 662 –^4
= 0.423 or 1.577 (to 3 d.p.).
EXAMPLE 1.33 Solve x^2 − 2 x + 2 = 0.
SOLUTION
The first thing to notice is that this cannot be factorised. The only two whole
numbers which multiply to give 2 are 2 and 1 (or −2 and −1) and they cannot be
added to get −2.
Comparing x^2 − 2 x + 2 to the form ax^2 + bx + c = 0
gives a = 1, b = −2 and c = 2.