If a + b + c = p and ab + bc + ca = q ; find a^2+ b^2+ c^2. - Vidyadip

Question

If $a + b + c = p$ and $ab + bc + ca = q ;$ find $a^2+ b^2+ c^2.$

✓

Answer

We know that
$( a + b + c )^2 = a^2 + b^2 + c^2 + 2( ab + bc + ca ) .....(1)$
Given that, $a + b + c = p$ and $ab + bc + ca = q$
We need to find $a^2 + b^2 + c^2 :$
Substitute the values of $( ab + bc + ca )$ and $( a + b + c )$
in the identity $(1),$ we have
$(p)^2 = a^2 + b^2 + c^2 + 2q$
$\Rightarrow p^2 = a^2 + b^2 + c^2 + 2q$
$\Rightarrow a^2 + b^2 + c^2$
$= p^2 - 2q$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "[2 Mark Question Answer]" from Expansions (Including Substitution)→Expansions (Including Substitution) — all question types→MATHEMATICS STD 9 question papers→ICSE Board STD 9 question papers→ICSE Board question paper generator→

Similar questions

The length and the breadth of a rectangle are $6 \ cm$ and $4 \ cm$ respectively. Find the height of a triangle whose base is $6 \ cm$ and the area is $3$ times that of the rectangle.

Factorise:
$x^4+5 x^2+9$ [Hint. Add and subtract $x^2$ ]

A point $P$ lies on the $x-$axis and another point $Q$ lies on the $y-$axis.
If the abscissa of point $P$ is $-12$ and the ordinate of point $Q$ is $-16;$ calculate the length of line segment $\text{PQ}.$

The difference between two positive numbers is $5$ and the sum of their squares is $73.$ Find the product of these numbers.

Using the standard formulae, expand:
$(6-2 y+4 z)^2$

Factorise$ :a^3- 27$

The hearts of 60 patients were examined through X-ray and the observations obtained are given below :
Find the median.

Diameter of heart (in mm)	120	121	122	123	124	125
Number of patients	7	9	15	12	6	11

Without using trigonometric tables, evaluate.
$\frac{\cos ^2 34^{\circ}+\cos ^2 56^{\circ}}{\sin ^2 59^{\circ}+\sin ^2 31^{\circ}}$

Factorise : $a^3-27 b^3+2 a^2 b-6 a b^2$

Evaluate $: 3 \cos 80^\circ \operatorname{cosec} 10^\circ + 2 \sin 59^\circ \sec 31^\circ .$