All positive

Algebra Level 3

What is the smallest integer $k$ such that $(k-2) x^2 + 8x + k + 4$ is positive for all real $x$ ?

3 5 2 1

1 solution

Rishu Jaar
Nov 4, 2017

For a Quadratic equation to be always positive, coefficient of $x^2$ must be positive and its discriminant must be strictly negative.

$\large \color{#20A900}{\implies k - 2 > 0}\hspace{1cm}\ldots ineq.^n(1)$ $\large \color{#3D99F6}{\implies 64 - 4(k-2)(k+4) < 0} \hspace{1cm}\ldots ineq.^n(2)$ Simultaneously solving both inequations gives $\rightarrow$ $\large \color{#D61F06}{\implies k \in (4,\infty)}$ Thus the smallest integer value is $\large\color{#69047E}{\boxed{k=5}}$

Relevant Wikis : - Quadratic expression - Parabola

Could you please prove that discriminant must be strictly negative for a polynomial to be positive?

Aman thegreat - 3 years, 7 months ago

A proof is quite simple, btw i said that For a quadratic expression to be always positive, it must have a negative discriminant and a positive coefficient of $x^2$ .

Rishu Jaar - 3 years, 7 months ago

Ok please prove it

Aman thegreat - 3 years, 7 months ago

@Aman Thegreat – Read this and also this

Rishu Jaar - 3 years, 7 months ago

@Rishu Jaar – I hope all your basic doubts are cleared after reading them and i have updated my solution. Thank you.

Rishu Jaar - 3 years, 7 months ago

All positive

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