Topic 2 of 9 · Form 3
Quadratics, inequalities and polynomials
Completing the square: ax² + bx + c = a(x + p)² + q gives the vertex (−p, q). Discriminant b² − 4ac decides the number of real roots; conditions for a line to meet, be a tangent to, or miss a curve.
Notes for this topic 1 file, in order
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Quadratic functions
Completing the square: ax² + bx + c = a(x + p)² + q gives the vertex (−p, q). Discriminant b² − 4ac decides the number of real roots; conditions for a line to meet, be a tangent to, or miss a curve.
Worked example: Find k if y = kx − 3 is a tangent to y = x² + 1.
x² − kx + 4 = 0; b² − 4ac = 0 → k² = 16 → k = ±4
Quadratic inequalities
Find the critical values, sketch the parabola and read off the region.
Worked example: Solve x² − x − 12 > 0.
(x − 4)(x + 3) > 0 → x < −3 or x > 4
Polynomials
Remainder theorem (remainder p(a) when divided by x − a) and factor theorem; solving cubic equations by finding one factor and then dividing.
Worked example: Solve x³ − 7x + 6 = 0.
p(1) = 0 → (x − 1)(x² + x − 6) = (x − 1)(x − 2)(x + 3) → x = 1, 2, −3
Simultaneous equations
One linear and one non-linear equation: substitute and solve the resulting quadratic.
Check yourself
- Write x² + 6x + 4 in the form (x + a)² + b.
- Find the remainder when 2x³ + x − 5 is divided by x − 2.
- For what values of k does x² + kx + 9 = 0 have no real roots?
Show answers
- (x + 3)² − 5
- 16 + 2 − 5 = 13
- k² < 36 → −6 < k < 6
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