An algebra problem by Nika Glunchadze

Algebra Level 4

Find minimal value of z z , if z = x 2 + 2 x y + 3 y 2 + 2 x + 6 y + 4 z=x^2+2xy+3y^2+2x+6y+4 .


The answer is 1.

Nika Glunchadze by Nika Glunchadze , 41, Georgia

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2 solutions

James Pohadi
Jul 14, 2016

z = x 2 + 2 x y + 3 y 2 + 2 x + 6 y + 4 z = x 2 + ( 2 y + 2 ) x + 3 y 2 + 6 y + 4 1 \begin{aligned} z & =x^{2}+2xy+3y^{2}+2x+6y+4 \\ z & =x^{2}+(2y+2)x+3y^{2}+6y+4 \color{#D61F06}{{ }^{1}} \end{aligned}

1 \color{#D61F06}{{ }^{1}} The minimum value of z z happen when x = − 2 y + 2 2 = − ( y + 1 ) x=-\dfrac{2y+2}{2}=-(y+1)

x = − ( y + 1 ) ⟹ z = ( − ( y + 1 ) ) 2 + ( 2 y + 2 ) ( − ( y + 1 ) ) + 3 y 2 + 6 y + 4 z = y 2 + 2 y + 1 − 2 y 2 − 2 y − 2 y − 2 + 3 y 2 + 6 y + 4 z = 2 y 2 + 4 y + 3 2 \begin{aligned} x=-(y+1) \implies & z=(-(y+1))^{2}+(2y+2)(-(y+1))+3y^{2}+6y+4 \\ & z=y^{2}+2y+1-2y^{2}-2y-2y-2+3y^{2}+6y+4 \\ & z=2y^{2}+4y+3 \color{#D61F06}{{ }^{2}} \end{aligned}

2 \color{#D61F06}{{ }^{2}} The minimum value of z z happen when y = − 4 2.2 = − 1 y=-\dfrac{4}{2.2}=-1

y = − 1 ⟹ z = 2 ( − 1 ) 2 + 4 ( − 1 ) + 3 z = 2 − 4 + 3 z = 1 \begin{aligned} y=-1 \implies & z=2(-1)^{2}+4(-1)+3 \\ & z=2-4+3 \\ & \boxed{z=1} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Nika Glunchadze
May 1, 2016

We have z = ( x 2 + y 2 + 1 + 2 x y + 2 x + 2 y ) + ( 2 y 2 + 4 y + 2 ) + 1 = ( x + y + 1 ) 2 + 2 ( y + 1 ) 2 + 1 z=(x^2+y^2+1+2xy+2x+2y)+(2y^2+4y+2)+1=(x+y+1)^2+2(y+1)^2+1 , so the minimal value of z is 1.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

It would be better if you mention in the problem that x , y x, y are real numbers.

You haven't proved that equality can be achieved, i.e. both x + y + 1 x + y + 1 and y + 1 y + 1 can equal to zero at the same time. It can be easily proved though.

Pranshu Gaba - 5 years, 1 month ago

Why can't we write z as the sum of two squares .?and then find.ans is coming 10

Tarun B - 3 years, 8 months ago

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