Question

if a/b= b/c,prove that (a+b+c)(a-b+c)=(a²+b²+c²)​

Answer 1

Step-by-step explanation:

HERE a/b=b/c,

then ac=b^2 equation 1

now

RHS

(a+b+c)(a-b+c)

THEN

a^2-ab+ac+ab-b^2+bc+ac-bc+c^2

we know that ac=b^2

a^2-b^2+c^2+b^2+b^2+ac-ac+bc-bc

a^2+b^2+c^2

RHS=LHS

HENCE PROVED

Answer 2

Given :

• a/b = b/c

To Prove:

• (a+b+c)(a-b+c)=(a²+b²+c²)

Proof :

Let , a/b = b/c = k (say)

⟹ a = bk and b = ck

⟹ a = ck.k and b = ck .... (1)

⟹ a = ck² .... (2)

And b = ck

Now, L. H. S of the given result, =(a+b+c)(a-b+c)

=(ck² + ck + c) (ck² - ck + c) [U eq 1 and 2]

=c(k² + k + 1) c(k² - k + 1)

= c²(k² + 1 + k) (k² + 1 - k)

= c²[(k² + 1)² - k²]

= c² ( k⁴ + 2k² + 1 - k²)

= c²(k⁴ + k² + 1) ....(3)

And R. H. S of the given result,

= a² + b² + c²

= (ck²)² + (ck)² + c²

= c²k⁴ + c²k² + c²

= c² (k⁴ + k² + 1) ....(4)

From,equation (3) and (4),

It follows that, (a+b+c) (a-b+c)=(a²+b²+c²)