First slide

Theory of equations

Question

if $α, β$ are the roots of $a x^{2} + b x + c = 0; α + h, β + h$ are the roots of $p x^{2} + q x + r = 0;$ and $D_{1}, D_{2}$ the respective discriminants of these equations, then $D_{1} : D_{2} =$

Moderate

A

$\frac{a^{2}}{p^{2}}$
B

$\frac{b^{2}}{q^{2}}$
C

$\frac{c^{2}}{r^{2}}$
D

none of these

Solution

Let $A = α + h$ and $B = β + h .$ Then,
$\begin{array}{l} A - B = (α + h) - (β + h) = α - β \\ \Rightarrow (A - B)^{2} = (α - β)^{2} \\ \Rightarrow (A + B)^{2} - 4 A B = (α + β)^{2} - 4 α β \\ \Rightarrow \frac{b^{2}}{a^{2}} - \frac{4 c}{a} = \frac{q^{2}}{p^{2}} - \frac{4 r}{p} \\ \Rightarrow \frac{b^{2} - 4 a c}{a^{2}} = \frac{4^{2} - 4 r}{p^{2}} \Rightarrow \frac{D_{1}}{a^{2}} = \frac{D_{2}}{p^{2}} \Rightarrow \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}} \end{array}$

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