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Algebra Level 4

Find the sum of sum and product of all values of k k such that p ( x ) = x 2 + k x + k + 2 p(x) = x^2 + kx + k + 2 is a perfect square.

0 0 − 4 -4 k k 4 4

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Dev Sharma by Dev Sharma , 20, India

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2 solutions

Rohit Udaiwal
Dec 24, 2015

A quadratic equation of the form a x 2 + b x + c ax^2+bx+c is a perfect square if it's discriminant, b 2 − 4 a c b^2-4ac is 0.

Now if x 2 + k x + k + 2 x^2+kx+k+2 is a perfect square then k 2 − 4 ( k − 2 ) = 0 or k 2 − 4 k − 8 = 0 k^2-4(k-2)=0\quad \text{or}\quad k^2-4k-8=0 .Sum and product of roots can be Vieta's and is 4 4 and − 8 -8 respectivly.Therfore their sum is 4 − 8 = − 4 . 4-8=\boxed{-4}.

14 Helpful 0 Interesting 0 Brilliant 1 Confused

N I C E ! ! NICE!!

Dev Sharma - 5 years, 5 months ago

When I saw sum and product vieta come up, though I don't think this problem should be lvl 4

Mardokay Mosazghi - 5 years, 5 months ago

Amazing solution did it same way!!

Mardokay Mosazghi - 5 years, 5 months ago
Rishabh Jain
Dec 24, 2015

P(x)= ( x + k 2 ) 2 − k 2 4 + k + 2 (x+\frac{k}{2})^2 -\frac{k^2}{4}+k+2 . For P(x) to be a perfect square- k 2 4 + k + 2 \frac{k^2}{4}+k+2 =0 which gives k 2 − 4 k − 8 k^2 -4k-8 =0 sum and product of whose roots are 4 and -8 respectively,hence 4+(-8)=-4

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Did the same way

Ayush Sharma - 5 years, 5 months ago

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