Core Maths C1 Revision Notes - Mr Barton Maths

The discriminant b 2 – 4ac
In the formula for the quadratic equation
ax 2 + bx + c = 0 ⇒ x =
− b ±
2
b − 4ac
2a
i) there will be two distinct real roots if b 2 – 4ac > 0
ii) there will be only one real root if b 2 – 4ac = 0
iii) there will be no real roots if b 2 – 4ac < 0
Example: For what values of k does the equation 3x 2 – kx + 5 = 0 have
i) two distinct real roots,
ii) exactly one real root
iii) no real roots.
Solution: The discriminant b 2 – 4ac = (–k) 2 – 4 × 3 × 5 = k 2 – 60
⇒ i) for two distinct real roots k 2 – 60 > 0
⇒ k 2 > 60 ⇒ k < –√60, or k > +√60
and ii) for only one real root k 2 – 60 = 0
⇒ k = ±√60
and iii) for no real roots k 2 – 60 < 0
⇒ –√60 < k < +√60.
Miscellaneous quadratic equations
a) 2 sin 2 x – sin x – 1 = 0.
Put y = sin x to give 2y 2 – y – 1 = 0
⇒ (y – 1)(2y + 1) = 0 ⇒ y = 1 or y = –½
⇒ sin x = 1 or –½ ⇒ x = 90°, 210° or 330° from 0° to 360°.
b) 3 2x – 10 × 3 x + 9 = 0
Notice that 3 2x = (3 x ) 2 and put y = 3 x to give
y 2 – 10y + 9 = 0 ⇒ (y – 9)(y – 1) = 0
⇒ y = 9 or y = 1
⇒ 3 x = 9 or 3 x = 1 ⇒ x = 2 or x = 0.
c) y – 3√y + 2 = 0. Put √y = x to give
x 2 – 3x + 2 = 0 and solve to give x = 2 or 1
⇒ y = x 2 = 4 or 1.
C1 14/04/2013 SDB
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