Quadratic Expressions - I

Quadratic Expressions - I

12. Transformed Equations :Let \( \alpha ,\beta \) be the roots of f(x) = ax²+bx+c=0, then


Problem (Vii): The quadratic equation whose roots are 1 more than the roots of x²-2x+1=0 is 
Sol :    Let f(x) = x²-2x+1=0. The quadratic equation whose roots are one more than the roots of 
    f(x) =0 is f(x-1)=0
    \( \therefore \)(x-1)²-2 (x-1)+1=0
    \( \Rightarrow \)x²-2x+1-2x+2+1=0
    \( \Rightarrow \)x²-4x+4=0

Problem (viii) :The quadratic equation whose roots are the reciprocals of the roots of ax²+bx+c=0 is 
Sol:    Let f(x) = ax²+bx+c=0
The quadratic equation whose roots are reciprocals of the roots of 
\( f(x)\, = 0\,\,is\,\,f\left( {\frac{1} {x}} \right) = 0 \)
\( \therefore f\left( {\frac{1} {x}} \right) = a\left( {\frac{1} {x}} \right)^2 + b\left( {\frac{1} {x}} \right) + c = 0 \)
\( \, \Rightarrow \frac{a} {{x^2 }} + \frac{b} {x} + c = 0 \)
\( \Rightarrow a + bx + cx^2 = 0 \)
\( \Rightarrow cx^2 + bx + a = 0 \)

Problem (ix):    If the roots of the equation ax²+bx+c=0 are the reciprocals of the roots of the equation px²+qx+r=0, then the value of acq²-b²pr is 
Sol:    Two equations \( a_1 x^2 + b_1 x + c_1 = 0\,\,and\,\,\,a_2 x^2 + b_2 x + c_2 = 0 \) have same roots if and only if \( \frac{{a_1 }} {{a_2 }} = \frac{{b_1 }} {{b_2 }} = \frac{{c_1 }} {{c_2 }} \) We know, the equation whose roots are the reciprocals of the roots of the equations ax²+bx+c=0 is cx²+bx+a=0 
    but given it is px²+qx+r=0
 \( \therefore both\,cx^2 + bx + a = 0 \) and \( px^2 + qx + r = 0 \) represents the same equations
\( \therefore \,\,\,\,\,\frac{c} {p} = \frac{b} {q} = \frac{a} {r} \) \( \Rightarrow \)acq² -prb² =0
\( \therefore \,\,\,\frac{{ac}} {{pr}} = \frac{{b^2 }} {{q^2 }} \)