Directly prove that $2x^2 -4x + 3 > 0$ for all real $x$

Question

I'm asked (for homework which isn't graded but instead the basis of a quiz) to directly prove that $2x^2 -4x + 3 > 0$ for all real $x$.

I am VERY new to proofs.

The textbook's only example is a case that was simplified to ( foo )^2 + bar, and it was assumed since ( foo )^2 is always positive that ( foo )^2 must be grater than or equal to bar.

I don't see a way to simplify this into ( something )^2 + something else.

Do you remember the method of "completing the square" in order to solve quadratic equations? The idea of that is exactly what's needed here. — Semiclassical, Aug 23 '14 at 17:50
I didn't see this as a completing the square problem, totally caught me off guard, thanks guys! — Cody Smith, Aug 23 '14 at 17:58
You're not that new to proofs. There's no "hard line" between proofs and the exercices you've done all your mathematical life, the question "solve $5x+2=3$" is asking for a proof that a particular number solves that equation. — Jack M, Aug 23 '14 at 20:34

score 4 · Accepted Answer · 2014-08-23T21:51:34.553

4

Complete the square:

$$2x^2-4x+3=(\sqrt 2 x)^2-2\times(\sqrt2 x)\times\frac2{\sqrt2}+\left(\frac2{\sqrt2}\right)^2-\left(\frac2{\sqrt2}\right)^2+3\\=\left(\sqrt 2 x-\left(\frac2{\sqrt2}\right)\right)^2+1$$

edited Aug 23 '14 at 21:51

answered Aug 23 '14 at 17:52

$=\left(\sqrt 2 x\color{red}-\left(\frac2{\sqrt2}\right)\right)^2+1$ – mike Aug 23 '14 at 20:27

score 3 · Answer 2 · answered Aug 23 '14 at 17:52

$D = (-4)^2 - 4*3*2 < 0$, hence the polynomial $2x^2 - 4x + 3$ has no real roots, and so its graph does not intersect the horizontal axis. Thus, the value of this polynomial never change its sign, and, for, instance, when $x = 0$ it has the value $3 > 0$, so it always take positive values.

score 3 · Answer 3 · answered Aug 23 '14 at 20:30

3

This is a slight simplification of @Sami's solution:

$$2x^2-4x+3=2(x^2-2x+1)+1=2(x-1)^2+1>0$$

answered Aug 23 '14 at 20:30

mike

5,604

Directly prove that $2x^2 -4x + 3 > 0$ for all real $x$

3 Answers3