An algebra problem by The Tech Savvy

Algebra Level pending

x 4 + 12 x 3 + 46 x 2 + a x + b x^4 + 12 x^3 + 46 x^2 + ax + b is a square of a quadratic expression. Find ( a − b ) 2 (a-b)^2 .


The answer is 1225.

The Tech Savvy by The Tech Savvy , 19, India

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1 solution

Let the quadratic expression be x 2 + m x + n x^2+mx+n . Now, ( x 2 + m x + n ) 2 = x 4 + 2 m x 3 + ( m 2 + 2 n ) x 2 + 2 m x + n 2 (x^2+mx+n)^2=x^4+2mx^3+(m^2+2n)x^2+2mx+n^2 On comparing this expression with the given expression x 4 + 12 x 3 + 46 x 2 + a x + b x^4+12x^3+46x^2+ax+b we get, 2 m = 12 ⟹ m = 6 2m=12 \implies m=6 m 2 + 2 n = 46 ⟹ n = 5 m^2+2n=46 \implies n=5 2 m n = a ⟹ a = 60 2mn=a \implies a=60 n 2 = b ⟹ b = 25 n^2=b \implies b=25 Therefore, ( a − b ) 2 = ( 60 − 25 ) 2 = 1225 (a-b)^2=(60-25)^2=\boxed{1225}

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