Question

if p,q are the roots of ax^2+bx+c=0 then find
$( \frac{1}{p} - \frac{1}{q} ) {}^{2}$
I want urgently ​

Answer 1

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★ Sᴏʟᴜᴛɪᴏɴ :-

Given ,

• Quadratic equation → ax² + bx + c = 0
• p , q are the roofs of te given equation.

We need to find ,

• $\sf\left[\dfrac{1}{p} - \dfrac{1}{q}\right]^2$

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Simplifying , the given question we have

➥ $\sf \left[\dfrac{1}{p}-\dfrac{1}{q}\right]^2$

➥$\sf \left[\dfrac{q-p}{pq}\right]$

Using

• $\bf \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$

➥ $\sf\dfrac{(q-p)^2}{(pq)^2}$

Using

• (a - b)² = a² + b² - 2ab

➥ $\sf\dfrac{q^2+p^2-2pq}{(pq)^2}$

As we know that

• Sum of roots (p + q) = $\bf\dfrac{-b}{a}$

• Product of roots (pq) = $\bf\dfrac{c}{a}$

• p² + q² = $\bf \dfrac{b^2-2ac}{a^2}$

Substituting we have

➥ $\sf \dfrac{\frac{b^2-2ac}{a^2} + 2\left(\frac{c}{a}\right)}{\left(\frac{c}{a}\right)^2}$

➥ $\boxed{\bf \dfrac{b^2 - 4ac}{c^2}}$

Hence , (1/p- 1/q)² = $\boxed{\bf \dfrac{b^2 - 4ac}{c^2}}$

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