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Algebra Level 2

How many integer values of $u$ satisfy $u^4+u^3-12u^2<0?$

The answer is 5.

3 solutions

Akhil Bansal
Nov 24, 2015

$\large u^4 + u^3 - 12u^2 < 0$ $\large u^2(u^2 + u - 12) < 0$ $\large u^2(u-3)(u+4) < 0$ Solving inequality using number line method . $\large u \in (-4,0) \cup (0,3)$ $\large u \equiv \{ -3 , -2 , -1 , 1 , 2 \}$

It's called wavy curve method also

Ankur Verma - 4 years, 1 month ago

u e (x+3)(x-4) ???

Albnor Leku - 5 years, 6 months ago

(x+3)(x-4)=x2-x-12 and not x2+x-12

damien G - 5 years, 2 months ago

How comes that a 4 grade polynomial expression has 5 solutions?

Majo C O - 5 years, 6 months ago

Because that was not an equation, that's is an inequalaity and an inequality can have zero,one,... infinite solutions.

Ex. $x > 0$ has infinite solutions and $x^2 < 0$ has no solution.

Akhil Bansal - 5 years, 6 months ago

x^2 <0 has no REAL solutions

John Wyatt - 5 years, 6 months ago

Its an inequation. If a 4 degree polynomial is expressed as a different inequation for example if the same sum had a (>) sign instead of (<) sign, it would have had infitely many solutions.

Pravar Parekh - 5 years, 5 months ago

Alex Li
Nov 26, 2015

u^4 + u^3 - 12 * u^2 < 0

u^2 * (u^2 + u - 12) < 0

u^2 * (u+4) * (u-3) < 0

Therefore, the graph of u^4 + u^3 - 12 * u^2 intersects 3 times - at 0,-4, and 3.

Then just plug in points between -4, 0 and 0,3. You will find that they are both negative.

So, the solution is [-4,3], right? WRONG. -4,0, and 3 are all roots... meaning that they are all equal to 0. Since the inequality is less than or equal to, the solutions are all numbers between -4 and 3, except for 0.

Count them up and you get 5

Sanjwal Singhs
Nov 26, 2015

Akhil Bansal's solution is correct the fact that 0 can't be a solution has to be taken while solving using wavy curve . as f(0) is 0 and not less than 0

How do you count?

The answer is 5.

3 solutions

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