Question

If the roots of the quadratic equation p(q – r)x2 + q(r – p)X + r(p - q) = 0 are equal, show that I/p+1/r =2/q
​

Answer 1

Given :-

• A quadratic equation, p(q - r)x² + q(r - p)x + r(p - q) = 0, whose roots are equal.

To Prove :-

• 1/p + 1/r = 2/q

Solution :-

Comparing this given quadratic equation with the standard form of quadratic equation i.e., ax² + bx + c = 0 , we get

• a = p(q - r)
• b = q(r - p)
• c = r(p - q)

For real roots, the discriminant of a quadratic equation is 0, which is equal to b² - 4ac

Here,

⇒ b² - 4ac = 0

⇒ { q(r - p) }² - 4p(q - r)r(p - q) = 0

⇒ q²(r² + p² - 2pr) - 4(pq - pr)(pr - qr) = 0

⇒ q²(r² + p² - 2pr) = 4( p²qr - pq²r - p²r² + pqr²)

⇒ q²r² + q²p² - 2q²pr = 4p²qr - 4pq²r - 4p²r² + 4pqr² = 0

⇒ q²r² + q²p² - 2q²pr - 4p²qr + 4pq²r + 4p²r² - 4pqr² = 0

Which is equal to the identity of,

⇒ (pq + qr - 2pr)² = 0

⇒ pq + qr = 2pr

Divide both sides by pqr, we get

⇒ pq/pqr + qr/pqr = 2pr/pqr

⇒ 1/r + 1/p = 2/q

hence, Proved!

Answer 2

${\large{\pmb{\sf{\underline{Required\: Solution...}}}}}$

Given that: The roots of the quadratic equation p(q – r)x² + q(r – p)x + r(p - q) = 0 are equal.

To show: 1/p+1/r =2/q

Full Solution:

~ Firstly by using the general form of quadratic equation that is ax²+bx+c=0 we have to compare, on comparing we get the following results;?:

• a as p(q-r)
• b as q(r-p)
• c as r(p-q)

~ Now as it's given that p(q – r)x² + q(r – p)x + r(p - q) = 0 are equal, this tells us that here we have to use formula of discriminant that is b²-4ac=0. Now according to this we have to put the values. Let's do it!

➼ b²-4ac=0

➼ [q(r-p)]² - 4[p(q-r)r(p-q)] = 0

➼ q²(r²+p²-2pr) - 4p(q-r)r(p-q) = 0

➼ q²(r²+p²-2pr) - 4(pq-pr)(pr-qr) = 0

➼ q²(r²+p²-2pr) = 4(pq-pr)(pr-qr) = 0

➼ q²(r²+p²-2pr) = 4(p²qr - pq²r - p²r² + pqr²) = 0

➼ q²r² + q²p² - 2q²pr = 4p²qr - 4pq²r - 4p²r² + 4pq² = 0

~ Now we have to use the given identity:

➼ (pq-qr-2pr)² = 0

➼ pq-qr = 2pr

~ Now at last,

➼pq/pqr -qr/pqr = 2pr/pqr

• Henceforth, proved!

${\large{\pmb{\sf{\underline{Additional \; Knowledge...}}}}}$

Some knowledge about Quadratic Equations -

★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a

★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a

★ Discriminant is given by b²-4ac

Discriminant tell us about there are solution of a quadratic equation as no solution, one solution and two solutions.

★ A quadratic equation have 2 roots

★ ax² + bx + c = 0 is the general form of quadratic equation

★ D > 0 then roots are real and distinct.

★ D = 0 then roots are real and equal.

★ D < 0 then roots are imaginary.

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━