2.9.2 Quadratic Equation, SPM Practice (Paper 1)

Question 6:
Write and simplify the equation whose roots are the reciprocals of the roots of

3 x^{2} + 2 x - 1 = 0

, without solving the given equation.

Solution:

Question 7:
Find the value of p if one root of

x^{2} + p x + 8 = 0

is the square of the other.

Solution:

Question 8:
If one root of

2 x^{2} + p x + 9 = 0

is twice the other, find the values of p.

Solution:

Question 9:
The roots of the equation

6 x^{2} + h x + 1 = 0 are α and β, whereas 3 α and 3 β

are the roots of the equation

2 x^{2} - x + k = 0

. Find the value of h and k.

Solution:

\begin{array}{l} 6 x^{2} + h x + 1 = 0 \\ a = 6, b = h, c = 1 \\ Roots = α, β \\ sor : \\ α + β = - \frac{b}{a} \\ α + β = - \frac{h}{6} ……… (1) \\ p o r : \\ α β = \frac{c}{a} \\ α β = \frac{1}{6} ……… (2) \\ 2 x^{2} - x + k = 0 \\ a = 2, b = - 1, c = k \\ Roots = 3 α, 3 β \\ sor : \\ 3 α + 3 β = - \frac{b}{a} \\ 3 (α + β) = - \frac{(- 1)}{2} \\ α + β = \frac{1}{6} ……… (3) \\ p o r : \\ 3 α (3 β) = \frac{c}{a} \\ 9 α β = \frac{k}{2} \\ k = 18 α β ……… (4) \\ Substitute (3) into (1) \\ α + β = - \frac{h}{6} \\ \frac{1}{6} = - \frac{h}{6} \\ h = - 1 \\ Substitute (2) into (4) \\ k = 18 α β \\ k = 18 (\frac{1}{6}) \\ k = 3 \end{array}

Question 10:
Find the range of values of p for which the equation

2 x^{2} + 5 x + 3 - p = 0

has two real distinct roots.

Solution: