 # 2.10.1 Quadratic Equations, SPM Practice (Paper 2)

2.10.1 Quadratic Equations, SPM Practice (Paper 2)

Question 1:
(a)  Find the values of k if the equation (1 – k) x2– 2(k + 5)x + k + 4 = 0 has real and equal roots.
Hence, find the roots of the equation based on the values of k obtained.
(b)  Given the curve y = 5 + 4x x2 has tangen equation in the form y = px + 9. Calculate the possible values of p.

Solutions:
(a)
For equal roots,
b2 – 4ac = 0
[–2(k + 5)] 2 – 4(1 – k)( k + 4) = 0
4(k + 5) 2 – 4(1 – k)( k + 4) = 0
4(k2 + 10k + 25) – 4(4 – 3k k2) = 0
4k2 + 40k + 100 – 16 + 12k + 4k2 = 0
8k2 + 52k + 84 = 0
2k2 + 13k + 21 = 0
(2k + 7) (+ 3) = 0

If $k=-\frac{7}{2}$  , the equation is
$\begin{array}{l}\left(1+\frac{7}{2}\right){x}^{2}-2\left(-\frac{7}{2}+5\right)x-\frac{7}{2}+4=0\\ \frac{9}{2}{x}^{2}-3x+\frac{1}{2}=0\end{array}$

9x2 – 6x + 1 = 0
(3x – 1) (3x – 1) = 0
x =

If k = –3, the equation is
(1 + 3)x 2 – 2(–3 + 5)x – 3 + 4 = 0
4x2 – 4x + 1 = 0
(2x – 1) (2x – 1) = 0
x = ½

(b)
y = 5 + 4x x2 ----- (1)
y = px + 9 ---------- (2)
(1)  = (2), 5 + 4x x2= px + 9
x2 + px – 4x + 9 – 5 = 0
x2 + (p – 4)x + 4 = 0

Tangen equation only has one intersection point with equal roots.
b2 – 4ac = 0
(p – 4)2 – 4(1)(4) = 0
p2 – 8p + 16 – 16 = 0
p2 – 8p = 0
p (p – 8) = 0
Therefore, p = 0 and p = 8.

Question 2:
Given α and β are two roots of the quadratic equation (2x + 5)(x + 1) + p = 0 where αβ = 3 and p is a constant.
Find the value p, α and of β.

Solutions:
(2x + 5)(x + 1) + p = 0
2x2 + 2x + 5x + 5 + p = 0
2x2 + 7x + 5 + p = 0
*Compare with, x2– (sum of roots)x + product of roots = 0
Product of roots, αβ = 3
$\frac{5+p}{2}=3$
5 + p = 6
p = 1

Sum of roots = $-\frac{7}{2}$

2+ 6 = 7α ← (multiply both sides with 2α)
2+ 7α + 6 = 0
(2α + 3)(α + 2) = 0
2α + 3 = 0   or α + 2 = 0
α=− 3 2    α = –2

Substitute α = –2 into (3),